The Laws of Averages: Part 1, Fruit Salad

This essay is long-ish — and is best saved for a time when you have time to read it in its entirety. It will be worth the wait and the eventual effort. It comes in three sections: a Primer on Averages, a general discussion of Fruit Salad metrics, and a more in-depth discussion of an example using a published study.

NB: While this essay is using as an example a fairly recent study by Catherine M. O’Reilly, Sapna Sharma, Derek K. Gray, and Stephanie E. Hampton, titled “Rapid and highly variable warming of lake surface waters around the globe”[ .pdf here; poster here, AGU Meeting video presentation here ], it is not meant to be a critique of the paper itself — I will leave that to others with a more direct interest. My interest is in the logical and scientific errors, the informational errors, that can result from what I have playfully coined “The First Law of Averages”.

Averages: A Primer

As both the word and the concept “average” are subject to a great deal of confusion and misunderstanding in the general public and both word and concept have seen an overwhelming amount of “loose usage” even in scientific circles, not excluding peer-reviewed journal articles and scientific press releases, let’s have a quick primer (correctly pronounced “prim – er”), or refresher, on averages (the cognizanti can skip this bit and jump directly to Fruit Salad).

and, of course, the verb meaning to mathematically calculate an average, as in “to average”.

Since there are three major types of “averages” — the mode, the median, and the mean — a quick look at these:

Several of these definitions refer to “a set of data”… In mathematics, a set is a well-defined collection of distinct objects, considered as an object in its own right. (For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written {2,4,6}.)

This image summarizes the three different common “averages”:

Here we see the Ages at which patients develop Stage II Hypertension (severe HBP – high blood pressure) along the bottom (x-axis) and the Number of Patients along the left vertical axis (y-axis). This bar graph or histogram shows that some patients develop HBP fairly young, in their late 30 and 40s, after 45 the incidence increases more or less steadily with advancing age to peak in the mid-60s, falling off after that age. We see what is called a skewed distribution, skewed to the right. This shewdness (right or left) is typical of many real world distributions.

What we would normally call the average, the mean, calculated by adding together all the patient’s ages at which they developed HBP and dividing by the total number of patients, though mathematically correct, is not very clinically informative. While it is true that the Mean Age for Developing HPB is around 52 it is far more common to develop HPB in one’s late 50s to mid- 60s. There are medical reasons for this skewing of the data — but for our purposes, it is enough to know that those outlying patients who develop HPB at younger ages sk

ew the mean — ignoring the outliers at the left would bring the mean more in line with the actual incidence figures.

For the medically inclined, this histogram hints that there may be two different causes or disease paths for HPB, one that causes early onset HPB and one related to advancing age, sometimes known as late High Blood Pressure.

(In this example, the Median Age for HPB is not very informative at all.)

Our HPB example can be read as “Generally, one begins their real risk of developing late HPB in their mid-40s and the risk continues to increase until their mid-60s. If you haven’t developed HPB by 65 or so, your risk decreases with additional years, though you still must be vigilant.”

Different data sets have different information values for the different types of averages.

Housing prices for an area are often quoted as Median Housing Costs. If we looked at the mean, the average would be skewed upward by the homes preferred by the wealthiest 1% of the population, homes measured in millions of dollars (see here, and here, and here).

Stock markets are often judged by things like the Dow Jones Industrial Average (DJIA) [which is a price-weighted average of 30 significant stocks traded on the New York Stock Exchange (NYSE) and the NASDAQ and was invented by Charles Dow back in 1896]. A weighted average is a mean calculated by giving values in a data set more influence according to some attribute of the data. It is an average in which each quantity to be averaged is assigned a weight, and these weightings determine the relative importance of each quantity on the average. The S&P 500 is a stock market index tracks the 500 most widely held stocks on the New York Stock Exchange or NASDAQ. [A stock index … is a measurement of the value of a section of the stock market. It is computed from the prices of selected stocks, typically a weighted average.]

Family incomes are reported by the US Census Bureau annually as the Median Household Income for the United States [$55,775 in 2015].

Life Expectancy is reported by various international organizations as “average life expectancy at birth” (worldwide it was 71.0 years over the period 2010–2013). “Mathematically, life expectancy is the mean number of years of life remaining at a given age, assuming age-specific mortality rates remain at their most recently measured levels. … Moreover, because life expectancy is an average, a particular person may die many years before or many years after the “expected” survival.” (Wiki).

Using any of the major internet search engines to search phrases including the word “average” such as “average cost of a loaf of bread”, “average height of 12-year-old children” can keep one entertained for hours.

However, it is doubtful that you will be more knowledgeable as a result.

This series of essays is an attempt to answer this last point: Why studying averages might not make you more knowledgeable.

Fruit Salad

We are all familiar with the concept of comparing Apples and Oranges.

Sets to be averaged must be homogeneous, as in comparable and not so heterogeneous as to be incommensurable.

Problems arise, both physically and logically, when attempts are made to find “averages” of non-comparable or incommensurable objects — objects and/or measurements, which do not logically or physically (scientifically) belong in the same “set”.

The discussion of sets for Americans schooled in the 40s and 50s can be confusing, but later, younger Americans were exposed to the concepts of sets early on. For our purposes, we can use a simple definition of a collection of data regarding a number of similar, comparable, commensurable, homogeneous objects, and if a data set, the data being itself comparable and in compatible measurement units. (Many data sets contains many sub-sets of different information about the same set of objects. A data set about a study of Eastern Chipmunks might include sub-sets such as height, weight, estimated age, etc. The sub-sets must be internally homogeneous — as “all weights in grams”.)

One cannot average the weight and the taste of a basket of apples. Weight and taste are not commensurable values. Nor can one average the weight and color of bananas.

Likewise, one cannot logically average the height/length of a set like “all animals living in the contiguous North American continent (considered as USA, Canada, and Mexico)” Why? Besides the difficulty in collecting such a data set, even though one’s measurements might all be in centimeters (whole or fractional), “all animals” is not a logical set of objects when considering height/length. Such a set would include all animals from bison, moose and Kodiak bears down through cattle, deer, dogs, cats, raccoons, rodents, worms, insects of all descriptions, multi-cellular but microscopic animals, and single-celled animals. In our selected geographical area there are (very very roughly) an estimated one quintillion five hundred quadrillion (1,500,000,000,000,000,000) insects alone. There are only 500 million humans, 122 million cattle, 83 million pigs and 10 million sheep in the same area. Insects are small and many in number and some mammals are comparatively large but few in number. Uni- and multicellular microscopic animals? Each of the 500 million humans has, on average, over 100 trillion (100,000,000,000,000 ) microbes in and on their body. By any method — mean, median, or mode — the average height/length of all North American animals would be literally vanishing small — so small that “on average” you wouldn’t expect to be able to see any animals with unaided eyes.

To calculate an average of any type that will be physically, scientifically meaningful as well as logical and useful, the set being averaged must itself make sense as a comparable, commensurable, homogenous collection of objects with data about those objects being comparable and commensurable.

As I will discuss later, there are cases where the collection (the data set) seems proper and reasonable, the data about the collection seems to be measurements in comparable units and yet the resulting average turns out to be non-physical — it doesn’t make sense in terms of physics or logic.

These types of averages, of disparate, heterogeneous data sets — in which either the measurements or the objects themselves are incommensurable — like comparing Apples and Oranges and Bananas — give a results which can be labelled Fruit Salad and have applicability and meaning that ranges from very narrow through nonsensical to none at all.

“Climate Change Rapidly Warming World’s Lakes”

This is claimed as the major finding of a study by Catherine M. O’Reilly, Sapna Sharma, Derek K. Gray, and Stephanie E. Hampton, titled “Rapid and highly variable warming of lake surface waters around the globe”[ .pdf here; poster here, AGU Meeting video presentation here ]. It is notable that the study is a result of the Global Lake Temperature Collaboration (GLTC) which states: “These findings, the need for synthesis of in situ and remote sensing datasets, and continued recognition that global and regional climate change has important impacts on terrestrial and aquatic ecosystems are the motivation behind the Global Lake Temperature Collaboration.”

The AGU Press Release regarding this study begins thus: “Climate change is rapidly warming lakes around the world, threatening freshwater supplies and ecosystems, according to a new study spanning six continents.”

“The study, which was funded by NASA and the National Science Foundation, found lakes are warming an average of 0.61 degrees Fahrenheit (0.34 degrees Celsius) each decade. That’s greater than the warming rate of either the ocean or the atmosphere, and it can have profound effects, the scientists say.”

So, what is being measured and reported? Buried in the AGU Video presentation, Simon Hook, of JPL and one of the co-authors, in the Q&A session, reveals that “these are summertime nighttime surface temperatures.” Let me be even clearer on that — these are summertime nighttime skin surface water temperatures as in “The SST directly at the surface is called skin SST and can be significantly different from the bulk SST especially under weak winds and high amounts of incoming sunlight …. Satellite instruments that observe in the infrared part of the spectrum in principle measure skin SST.” [source] When pressed, Hook goes on to clarify that the temperatures in the study are greatly influenced by satellite measurement as the data is in large part satellite data, very little data is actually in situ [“in its original place or in position “ — by hand or buoy, for instance] measurements. This information is, of course, available to those who read the full study and carefully go through the supplemental information and data sets — but it is obscured by the reliance on stating, repeatedly “lakes are warming an average of 0.61 degrees Fahrenheit (0.34 degrees Celsius) each decade.“

What kind of average? Apples and Oranges and Bananas. Fruit Salad.

Here is the study’s map of the lakes studied:

One does not need to be a lake expert to recognize that these lakes range from the Great Lakes of North America and Lake Tanganyika in Africa to Lake Tahoe in the Sierra Nevada Mountains on the border of California and Nevada. Some lakes are smaller and shallow, some lakes are huge and deep, some lakes are in the Arctic and some are in the deserts, some lakes are covered by ice much of the year and some lakes are never iced over, some lakes are fed from melting snow and some are feed by slow-moving equatorial rivers.

Naturally, we would assume, that like Land Surface Temperature and Sea Surface Temperature, the Lake Water Temperature average in this study is weighted by lake surface area. No, it is not. Each lake in the study is given equal value, no matter how small or large, how deep or how shallow, snow fed or river fed. Since the vast majority of the study’s data is from satellite observations, the lakes are all “larger”, small lakes, like the reservoir for my town water supply, are not readily discerned by satellite.

So what do we have when we “average” the [summertime nighttime skin surface] water temperature of 235 heterogeneous lakes? We get a Fruit Salad — a metric that is mathematically correct, but physically and logically flawed beyond any use [except for propaganda purposes].

This is freely admitted in the conclusion of the study, which we can look at piecemeal: [quoted Conclusion in italics]

“The high level of spatial heterogeneity in lake warming rates found in this study runs counter to the common assumption of general regional coherence.”

Lakes are not regionally responding to a single cause — such as “global warming”. Lakes near one another or in a defined environmental region are not necessarily warming in similar manners or for the same reason, and some neighboring lakes have opposite signs of temperature change. The study refutes the researcher’s expectation that regional surface air temperature warming would correspond to regional lake warming. Not so.

Lakes are warming for geomorphic (having to do with form of the landscape and other natural features of the Earth’s surface) and local climate — not regionally, but individually. This heterogeneity implies lack of a single or even similar causes within regions. Lack of heterogeneity means that these lakes should not be consider a single set and thus should not be averaged together to find a mean.

Globally, lakes are not a physically meaningful set in the context of surface water temperature.

“The heterogeneity in surface warming rates underscores the importance of considering interactions among climate and geomorphic factors that are driving lake responses and prevents simple statements about surface water trends; one cannot assume that any individual lake has warmed concurrently with air temperature, for example, or that all lakes in a region are warming similarly.”

Again, their conclusion is that, globally, lakes are not a physically meaningful set in the context of surface water temperature yet they insist on finding a simple average, the mean, and basing conclusions and warnings on that mean.

“Predicting future responses of lake ecosystems to climate change relies upon identifying and understanding the nature of such interactions.”

The surprising conclusion shows that if they want to find out what is affecting the temperature of any given lake, they will have to study that lake and its local ecosystem for the causes of any change.

A brave attempt has been made at saving this study with ad hoc conclusions — but most are simply admitting that their original hypothesis of “Global Warming Causes Global Lake Warming” was invalidated. Lakes (at least Summertime Nighttime Lake Skin Surface Temperatures) may be warming, but they are not warming even in step with air temperatures, not reliably in step with any other particular geomorphic or climatic factor, and not necessarily warming even if air temperatures in the locality are rising. As a necessary outcome, they fall back on the “average” lake warming metric.

This study is a good example of what happens when scientists attempt to find the averages of things that are dissimilar — so dissimilar that they do not belong in the same “set”. One can do it mathematically — all the numbers are at least in the same units of degrees C or F — but such averaging gives results that are non-physical and nonsensical — a Fruit Salad resulting from the attempt to average Apples and Oranges and Bananas.

Moreover, Fruit Salad averages not only can lead us astray on a topic but they obscure more information than they illuminate, as is clearly shown by comparing the simplistic Press Release statement “lakes are warming an average of 0.61 degrees Fahrenheit (0.34 degrees Celsius) each decade” to the actual, more scientifically valid findings of the study which show that each lake’s temperature is changing due to local, sometimes even individual, geomorphic and climate causes specific to each lake and casting doubt on the idea of global or regional causes.

Another example of a Fruit Salad metric was shown in my long-ago essay Baked Alaska? which highlighted the logical and scientific error of averaging temperatures for Alaska as a single unit, the “State of Alaska”, a political division, when Alaska, which is very large, consists of 13 distinct differing climate regions, which have been warming and cooling at different rates (and obviously with different signs) over differing time periods. These important details are all lost, obscured, by the State Average.

Bottom Line:

It is not enough to correctly mathematically calculate the average of a data set.

It is not enough to be able to defend the methods your Team uses to calculate the [more-often-abused-than-not] Global Averages of data sets.

Data sets must be homogeneous, physically and logically. They must be data sets of like-with-like, not apples-and-oranges. Data sets, even when averages can be calculated with defensible methods, must have plausible meaning, both physically and logically.

Careful critical thinkers will be on the alert for numbers which, though the results of simple addition and division, are in fact Fruit Salad metrics, with little or no real meaning or with meanings far different than the ones claimed for them.

Great care must be taken before accepting that any number presented as an average actually represents the idea being claimed for it. Averages most often have very narrow applicability, as they obscure the details that often reveal the much-more-important actuality [which is the topic of the next essay in this series].

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Note on LOTI, HadCRUT4, etc.: It is my personal opinion that all combined Land and Sea Surface Temperature metrics, by all their various names, including those represented as indexes, anomalies and ‘predictions of least error’, are just this sort of Fruit Salad average. In physics if not Climate Science, temperature change is an indicator of change in thermal energy of an object (such as of a particular volume of air or sea water). In order to calculate a valid average of mixed air and water temperatures, the data set must first be equal units for equivalent volumes of same material (which automatically excludes all data sets of sea surface skin temperatures, which are volume-less). The temperatures of different volumes of different materials, even air with differing humidity and density, cannot be validly averaged without being converted into a set of temperature-equivalent-units of thermal energy for that material by volume. Air and water (and stone and road surfaces and plowed fields) have much different specific heat capacities thus a 1 °C temperature change of equal volumes of these differing materials represents greatly differing changes in thermal energy. Sea Surface (skin or bulk) Temperatures cannot be averaged with Surface Air Temperatures to produce a physically correct representation claimed as a change in thermal (heat) energy — the two data sets are incommensurable and such averages are Fruit Salad.

And yet, we see every day, these surface temperature metrics represented in exactly that non-physical way — as if they are quantitative proof of increasing or decreasing energy retention of the Earth climate system. This does not mean that correctly measured air temperatures at 2 meters above the surface and surface sea water temperatures (bulk — such as Argo floats at specific depths) cannot tell us something, but we must be very careful in our claims as to what they tell us. Separate averages of these data sets individually are nonetheless still subject to all the pitfalls and qualifications being presented in this series of essays.

“The global temperature exists. It has a precise physical meaning. It’s this meaning that allows us to say…

The LIA was cooler than today…it’s the meaning that allows us to say the day side of the planet is warmer than the nightside…The same meaning that allows us to say Pluto is cooler than earth and mercury is warmer.”

I must say I agree with his statement — and if Climate Scientists would limit its claims for various Global Temperature averages to these three concepts, their claims would be far more scientifically correct.

NB: I do not think it is correct to say “It has a precise physical meaning.” It may have a precise description but what it means for the Earth’s climate is far from certain and does not approach precise by any measure.

I expect opinions may vary on this issue.

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Author’s Comment Policy:

I am always anxious to read your ideas, opinions, and to answer your questions about the subject of the essay, which in this case is Fruit Salad Averages as defined above.

As regular visitors know, I do not respond to Climate Warrior comments from either side of the Great Climate Divide — feel free to leave your mandatory talking points but do not expect a response from me.

I am interested in examples of Fruit Salad Averages from the diverse range of science and engineering fields in which WUWT readers work.

It is a classic case of ‘apples and oranges’. If you take the average of an apple and an orange, the answer is a fruit salad. It is not a useful quantity for physics based calculations such as earth energy budget and the impact of a radiative “forcings”.

Roger Knights ==> Cruel Roger, to play on my weakness with semi-colons….only my Vassar educated editor understands the darkness that surrounds these vile creatures….
Truthfully, I write as if I were speaking — which sometimes plays havoc with punctuation-according-to-Style-Guides.
You are probably exactly right —

Why would neighbouring lakes have different? Maybe one is covered by algae.

My guess is that the researchers gathered the data, couldn’t find anything useful, and thrashed about with different methods of analysis until they found a ‘significant’ result. It’s similar to the dark chocolate hoax. This XKCD cartoon also explains the phenomenon.

The researchers did a bunch of work. They can’t say they didn’t find anything because nobody will publish that. The result is there for everyone to see.

commie ==> The study is from “…the Global Lake Temperature Collaboration (GLTC) which states: “These findings, the need for synthesis of in situ and remote sensing datasets, and continued recognition that global and regional climate change has important impacts on terrestrial and aquatic ecosystems are the motivation behind the Global Lake Temperature Collaboration.” ”

Nearly all of the data in the study is from satellite data, as I point out.

That XKCD cartoon should cause blushing among a few self proclaimed scientists. If they understand it, that is.

On the other hand, the dark chocolate hoax brings another problem to mind. What if fabricated temperature records showed an increase over the time the current 20 somethings grew taller? The correlation between fabricated temperatures and their average height is quite striking. There is even a pause.

Imagine if somebody started a hoax about that. Then politically motivated types could get advice for them to stop breathing out. And there could be legislation and conferences. The science is settled!

This is where a principle component analysis would come in handy. If you do it right, you feed everything into it, and it tells you what the important modes of variation are. In the case of global temperature measurements, it would probably make the UHI effect and data tampering pop right out, but you would also get to see how things vary with the season.

I didn’t know it had a name but, yes, it does. link Scientists admit there’s a problem with science but then you get folks like Neil deGrasse Tyson telling us that we have to accept the authority of science.

“Why would neighbouring lakes have different? Maybe one is covered by algae.”

I don’t know, but consider these NWS reports from Lake Champlain (One of their study lakes) yesterday. The temps are daytime and near shore. Yesterday was a clear, sunny, coolish day so clouds probably aren’t a factor.

Note: I don’t know where the USGS gauge at Burlington is located, but it’s unlikely to be even a kilometer from the gauge at the King St Ferry Dock. And Colchester Reef is only 12-15km NorthWest of Burlington. I’m guessing that the Burlington gauge is busted or was misread.

Although you finally get to it at the end, even your assessment method for defining Fruit Salad Average (FSA) is your own opinion. The people who create metrics such as “Global Mean Temperarure” have a different opinion, obviously. Thus we’ve really gained no ground here.

My foray into global warming skepticism falls along your reasonings, not to mention the act of measurement itself. But many many people in the sciences feel the methods of collection and interpretation are perfectly valid. If they weren’t, this all would’ve collapsed long ago.

I wonder – if we averaged a set composed of ‘dog’s breakfasts’ and ‘dog’s dinners’ could we outline the parameters of a ‘dog’s brunch’? We would have to make sure that all the dogs used the term ‘dinner’ to denote a midday repast. Additionally, does the use of one or the other of these expressions bear any correspondence to the user’s regional or social stratum origin? Anybody got a blank grant application that I could use?

Since forever we have kept horses, and dogs, the phrase instantly conjured images of my dogs (we loved them all!) happily banging out the doggie door first thing in the morning to make a cursory fence patrol then settling into the horse paddock for breakfast.

While “many, many people in science FEEL [their] methods of collection and interpretation are perfectly valid” doesn’t mean they are valid. Kip’s review/ analysis and lesson is pretty darn good and not just his opinion. I have had to deal with more than one “fruit salad” submitted paper or analysis in my career. We were using the work to make regulations that would dramatically affect lots of people’s livelihoods. The authors would argue vehemently that their analysis was valid. Often it took very little to demonstrate how invalid their “averaging” was.

tadchem, Clyde, and Kokoda ==> Cheers my heart to know that others read carefully and pay attention to the words.
My editor was again unavailable yesterday — thus a few of these little typos have slipped through.

Regarding the ‘meaning’ of a global average temperature, in the geophysical and the ecological senses of reality it is meaningless. No physical feature and no biological organism experiences an *average* temperature. All objects experience the local, transient temperature at their immediate location. There is no uniformity. There are only varying degrees of variability. The mesquite bush outside my office window can experience a minute change of temperature from minute to minute as clouds pass over, a small change from hour to hour as storms come and go, a larger change from night to day, and a tremendous change throughout the year. The bush has adapted to these changes and survives them all, and the ‘average’ temperature is irrelevant. It can survive temperatures over 100°F (as it did yesterday) and temperatures under 20° F (as it did 6 months ago). The pertinent life-threatening changes in its environment involve many other factors such as insects, browsing animals, wildfire, flash floods, etc.

Yes. One can imagine the limited utility of a number said to be the “average summer temperature” of Honolulu, or the even more limited utility of the “average summer temperature” of Denver; but the “average” of those two numbers has lost all utility. It describes nothing useful.

Thanks Kip – a very good start.
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Many years ago as sensors and algorithms (S&A) were being introduced to environmental studies, I wrote an essay for my instructor that included a look to the future.
Prior to S&A, land cover studies involved field excursions – usually students with clipboards criss-crossing an area and writing things like “block A has Pine trees, block B is a grass field, block C is the 3rd fairway.”
During the introduction of S&A, automated reports would be generated and then a “ground-truth” field trip would be conducted to see if the algorithms did, if fact, distinguish between the land cover types.
My paper’s hypothesis was that reliance on satellites to do the work of poorly paid graduate students was not necessarily a good thing.

[PS: On one “ground truth” trip, while a student was intent on recording land cover, a dog walked up and peed on his leg.]

John F. Hultquist == Anthony Watts’ Surface Station Project was that sort of “ground truthing” about the physical circumstances surrounding individual weather stations. I did a report on the station In Santo Domingo, Dominican Republic. (I did not get peed on).

Kip, the pronunciation is Prime- er, meaning basic or first. Children go to a primary school, not a primmery school. They are prime numbers, not primm numbers; the primary reason, not primmery reason. Jodie Foster couldn’t pronounce it either.

In my traditional English education during the 50s and 60s there is only one pronunciation, and it is not primm-er.

(I say traditional because it was long before the modern ‘progressive’ education systems that produce the high number of illiterates we now have in the UK and which have brought about the significant drop in standards of UK education when measured against international standards)

Obviously only a matter of varied usage, semi-dialect. In NZ it happens to be primmer just for the 1st 4 classes, 2yrs where we learnt to read and write; and become at least partly-civilised. Still spelt primer, though.

I agree completely with you that there is an overabundance of ‘Fruit Salad’ in climatology papers. I have complained previously here that averaging sea surface temperatures with land air temperatures distorts what is happening. Similarly, I have advocated reporting the global average of diurnal highs and lows individually, rather than a grand average, because there are different processes at work controlling the highs and lows and, again, those processes are hidden by a grand average. Further, I have suggested that land averages be grouped by climatic zones to see if all are responding similarly, which I believe they are not. It is surprising to me just how poor the analysis of temperature data has been, and that so few people have remarked that “the emperor has no clothes.”

With the current data quality and infill (guessing) your proposed analysis is likely impossible. We would have to deploy something similar to the Climate Reference Network sites to all the areas considered representative and then wait a century to have enough data to do the analysis. People today don’t seem to have that sort of patience, especially with chicken littles out there proclaim thermogeddon in the next 50 years.

OweninGA,
There is no question that the global temperature data set was never intended for the use that climatologists are trying to apply it to. However, we do currently have some who are using the sparsely-monitored Arctic temperatures to make broad statements about what is happening in the Arctic. I’m suggesting that those who are the ‘professionals’ in the field should be similarly looking at the other climate zones. The data available may not be optimal, but they might still provide some insights, if someone bothered to look. That effort alone might help to provide specifications for what an optimal climate monitoring network might need. Yes, the mentality seems to be one of “Ready, fire, aim.”

Kip, I had previously commented in other Threads about the absurd “averaging” of SST and LST’s. Additionally, it is passing strange that we find nothing wrong with “averaging” Arctic and Equatorial temperatures; differences in humidity, height of the tropopause, etc.

Dave ==> No one goes to this much work without a specific purpose in mind. The specific purpose of measuring/deriving the Earth’s Mean Surface Temperature, in the political climate today, it to prove that CO2 concentrations increasing in the atmosphere are causing to Earth system to retain extra energy incoming from the Sun (and that this is somehow dangerous instead of beneficial).

Since Surface Temperature, regardless of how calculated, does not represent the amount of energy retained (or not retained) it is either just interesting for its own sake (for the curious) or a fool’s errand.

The Global Average temperature and GCM’s are a construct. Maybe even a useful construct but still a “fruit salad” and casting of bones to attempt to foresee the future. What the results of this averaging exercise means will only be known in a few hundred years when folks look back on this period of time with the knowledge of hindsight.

An example that immediately comes to mind is the so-called “pay gap” between men and women. Claims of a significant difference in pay are based on averaging all salaries of men and all salaries of women. This averaging method does not consider factors such as different jobs (comparing an oil field worker with a fast food worker), different levels of experience, and different hours worked (full-time vs. part-time). No significant gap is found when comparing salaries of men and women with the exact same job, who work the exact same hours, who have the same levels of education and experience, etc.

More social science and political than natural or hard science, but it is a good example of how how to lie with statistics. It is also a good example of how entrenched false claims can become, even when debunked by disparate sources. Even when people know the truth, they may still promote fruit salad based conclusions that “support” their narrative (noble cause corruption). Example: the Department of Labor has tweeted the false statistic as fact, even though a) they should know better, and b) they do know better, as evidenced by their own website’s refutation.

Ally ==> Yes, lumping “men” and “women” into two categories and averaging each for comparison is a Fruit Salad exercise. I have seen better analyses of the gap — the gap gets smaller with closer attention to avoiding Fruit Salad — but may still exist as a societal artifact.

A Fruit Salad, indeed – and for the same reason as trying to “average” a global, or national, or State temperature. There are economic “microclimates” that vary just as much as meteorological ones.

I was very well paid as a “Senior Application Developer” – for my area. Just up the road, less than 90 miles away, I would have been somewhat underpaid – and darn near poverty level if I had been making that much and living in San Francisco, New York, etc.

Very good argument on a basic point. Where I was raised in Northern California, much of the influence on temperature was just how close one was to the ocean, and how many mountains were between you and the ocean. It could be 64F at the coast, 80F in the Santa Clara Valley (one range of mountains), and 95F in the Central Valley (two ranges of mountains). An average temperature would not mean much.

In your first figure you show the mean to be 52. This is the mean of the ages at which at least one patient developed HPB, not the mean of all the patients ages at which they developed HPB. That latter mean would be 57.38 if I read the chart correctly and is actually the more normal meaning of mean.

David Kleppinger ==> The HBP chart is taken from a web search for charts about averages. The MEAN is the mathematical mean of age of all patients added together divided by number of patients. The MEDIAN is the age at which there are as many patients above this age as below this age. The MODE is the age with the highest number of patients.
I can not vouch for the medical accuracy of the chart — not my data.

I’m just saying that was a bad chart to show as an example.
What that chart is showing as the mean appears to be the high value + low value / 2.
If you calculate it, the mean of the all the patient ages is 57.4, the mean of the ages (without considering the number of patients at that age) is 53.2.

David ==> You have a very good eye, sir. There is something amiss (in the numbers). It is quite possible that the chart does not show all the outliers at either end, thus making the mean and median appear to be in the wrong places. (again, not my chart)
The concepts in the essay are correctly stated — only a maths or stats eye would catch what you have caught!

“The global temperature exists. It has a precise physical meaning.” – agreed, but we haven’t been measuring the true global temperature, and I’m not sure we can. The true global temperature would be to capture all global temperatures at the same point in time and then average them. And then we need this same capture to happen for each unique surface area that’s in sunlight. If there is a time variance of when each temperature value included in the average is taken then we haven’t achieved the ‘precise physical meaning’ of a global temperature average.

“allows us to say the day side of the planet is warmer than the night side” – it does? Does it allow us to determine how the hemispheres compare? A single ‘global temperature’ value does not allow us to do any of that.

A ‘precise’ global temperature value would allow us to determine if the Earth is warming at the rate of 0.0000023 C degrees per hour to prove that we’ll have 2 C degrees warming in 100 years. I don’t believe we have that level of precision. And since we don’t, any proxy average global temperature value needs to have an error margin included.

To me bigger question is the meaning of the average of a non-linear quantity , temperature , rather than their 4th power , energy — in which computations are linear . That is , is the most meaningful average temps 4. ^ avg .25 ^
rather than temps avg
where : avg dup +/ swap % ;

Where I’ve taken the liberty of expressing the computations in CoSy‘s RPN Forth syntax and left out the StefanBoltzmann constant which drops out anyway because I think the difference in the computations is easily understood however expressed .

Bob Armstrong ==> It gets complicated….
The CO2 Global Warming hypothesis is that increasing concentrations of CO2 will cause the Earth system to retain additional energy (until it eventually regains equilibrium). As some part of this energy expresses as sensible heat, they measure temperatures. Temperatures of the atmosphere and seas are not proven to be reliable measures of retained energy — energy is not even being measured (with the exception of incoming and outgoing radiation). no attempt has been made to determine where that energy goes and how it is stored.
One might ask this question: when energy arrives from the Sun, at the Earth’s surface, what energy transformation take place? Some is stored as sensible heat, some is transformed into stored chemical energy, some moves the air around, some moves the seas, some is transformed into the energy potential of water moved from sea level to the tops of mountain ranges, etc etc.

I have recently become absolutely certain that the Al Gore , James Hansen GHG spectral “heat trapping” paradigm is false on the most basic physical level .

Gravity is left out of the equations and it cannot be . I have come to view this as “settled” despite it being apparently outside the ken of the “climate science” community — altho more here are realizing it .

If light is “heated” by gravity , it is only a matter of working out the equations to calculate the heating of all matter including atmospheres as one descends into a gravitational well .
—
That was somewhat of an interjection of my current state of understanding . The point I raised is a straight forward mathematical one . The general computation which leads to the 255K meme , a specific case I include in this slide ,
assumes a point or , in the more common derivation a flat disk . Thus whether the average is performed over temperatures or energies and then converted to temperatures makes no difference . However , the implicit assumption in the standard ” disk % 4 ” computation is in terms of energy .

( Again , excuse the crudeness of the CoSy notation compared to a polished APL due to it executing right at the x86 register level , but if anybody appreciates what’s being done , definitely check out CoSy . )

As you can see , in this example you get either 285.00 or 288.23 depending on whether you average temperature or average kinetic energy density ( power ) .

Thank you for a peek behind the mystical statistical curtain. Let me offer a point of clarification with your description of the HBP figure:
” — but for our purposes, it is enough to know that those outlying patients who develop HPB at younger ages skew the mean — ignoring the outliers at the left would bring the mean more in line with the actual incidence figures.”
The reality of the distribution is indeed negatively skewed, but it is incorrect to arbitrarily lop off the “outliers” to create a classic symmetric distribution with zero skew. If you can justify removing the inconvenient outliers like those on the left, there are statistical methods such as employing Chauvenet’s Criterion:http://www.statisticshowto.com/chauvenets-criterion/
Some might consider this as putting a mathematical thumb on the scales. Others might consider this as an excuse to clean up messy data instead of doing a better experiment.

Neil ==> The alternative, and this is part of the next essay, is to realize that averaging (in any way) obscures data. The histogram shows clearly that there seems to be two bumps — an earlier onset bump and a late-onset bump. This makes sense in medicine — there are likely to be differing causes (physical medical phenomena – disease pathways) for the two bumps.

In real life you frequently find outliers – spikes of unusually high numbers at either end of the normal distribution. When analysing statistics, these are sometimes just discarded. Other times, non-parametric methods are used to analyse them.

Another fruit salad is the average of immigrants vs local Inhabitants. Say that the average is 0.3% said nothing about a town of 120 (one hundred twenty) Inhabitants with more than 1000 (one thousand) of immigrant. A real situation here in Italy:

I live in Vicenza (close to Venice) and the data came from local TV. If you are interested I can check out the name of the city. But what is important here is the meaning of mean, not where this happen.

Sorry, I misunderstood the question. I supposed “they” was referred to the data but I see now that maybe it referred to “immigrants”, anyways this doesn’t matter because we are speaking about fruit-salad data not immigrants.

marianomarini ==> Yes, your last is correct — where do the immigrants come from was my question — but I was just curious as to who would be immigrating to Italy in the first place.
And you are also correct with this “this doesn’t matter because we are speaking about fruit-salad data not immigrants.”.

I hope this is related in my Project Management Experience. My company does this (incorrectly IMO) when calculation GP (Gross Profit) of any particular project. Emphasis is placed on total GP by combining material & labor input ‘costs’ and subtracting them from total contract ‘value’, then dividing to get a percentage (i.e. $2000 value – $1000 labor/material cost = $1000 profit (50%) – Everyone is happy we make 50% profit all the time. The problem with this is combining material costs and labor and equating them both to a dollar amounts. Sure it can be useful as an indicator but it hides the weighting each one provides. Like your example of comparing two lakes of different sizes/depths, what lies beneath the surface is what matters.

I have argued, by combing these different matrix values into one, the output does not provide meaningful information that helps drive decisions. If applying paint onto the equipment, one paint system costs $1000 and takes 32 labor units (say $3200) to apply. Another paint system costs $4000 and only 2 labor units ($200) to apply. Both paint systems cost $4200 to apply but of course the paint system that only took two hours to apply is superior from a productivity standpoint but on paper they become equal when combined.

They also omit another very important variable – linear time. Did I spend 4 hours in one day to make $1000 or did I spend one hour a week over a month (4 hours total) to make $1000. Both on paper make the same amount but have very different implications. I have argued a weighted duration factor needs to be incorporated into our costing model to no avail.

Of course other companies do this analysis much better but not mine. The law of averages with apples and oranges does not apply, the final number then becomes meaningless to make educated decisions/conclusions.

Duncan ==> Lots of interesting ways to misuse averages == illogically combining ‘objects’ is quite common. The misunderstanding seems to be that if one can get them into the same units (degrees, dollars, cm) then averaging them makes sense. That is simply not true,
Hope you can get through to your superiors.
The use of such averages is often a way to avoid having to look at more complicated data which might provide more insight.

Hence the famous saying, “lies, damn lies, and statistics.” Any time I see arithmetical mean being thrown around I think grade school science, yet this grade school science permeates government sponsored green science.

One horrible example I’ve dealt with personally involved the lesser prairie chicken and the pseudoscience of “avoidance behavior.” The FWS used junk science from papers that determined “mean avoidance behavior” of prairie chickens to infrastructure. Basically, they captured the chickens, put tracking beacons on them, and then tracked their movements for a few days. They then selected a point, i.e. a pump jack or wind turbine, and then calculated the mean distance of the bird’s position from this point, and called it mean avoidance. These means were then used to determine buffer zones around certain infrastructure that was purportedly then considered to be completely destroyed prairie chicken habitat, mitigation fees of up to $100,000 per acre were assessed, and then that money was used to bribe land owners into joining a conservation program. Of course, you could search the literature and find myriad reasons why the lesser prairie chicken were in decline, and each one treated equally as pertinent except for the two reasons that could not be extorted for money, drought and booming Asian Pheasant populations.

Most people basically saw this as two things: a way to extort industry (ones that actually create wealth) out of money to be funneled into the sue and settle green groups (most of these people are attorneys collecting HUGE paychecks from these settlements) and a way to fulfill a mandate of Agenda 21 — forcing people out of rural areas and into the cities. Luckily a federal judge in Texas saw it the same way and shut it down.

noaa ==> Not only different means — but each has a valid application which is narrow and often specific in place, time, range, etc. Modern usage is to consider “because it is an average” it application is universal — when just the reverse is true.

Even more to the point (as what I think noaaprogrammer is alluding to), is the use of standard deviation across time series where a single entry at one point in time may be close to the ‘mean’ but at a different time may be the outlier (such as the 2012 arctic sea ice extent which was right on the 1981-2010 mean in April but was an extreme outlier in September of that year). Lots of very good maths being applied in ways that maybe “I do not think it means what you think it means”.

steverichards1984 ==> As you probably already know, satellite-based instruments do not actually measure temperatures. The derive a “skin” temperature from sensor data.

Satellite instruments that observe in the infrared part of the spectrum in principle measure skin SST. One such instrument is the Advanced Very-High Resolution Radiometer (AVHRR, Schluessel et al. 1987) used on satellites operated by the National Oceanic and Atmospheric Administration (NOAA). In practice, AVHRR measurements have been tuned to bulk SST measurements made by buoys (May et al. 1998). We have worked to retune the AVHRR observations to skin SSTs that have been adjusted from bulk measurements from buoys across the globe. The new AVHRR skin SSTs have a nearly zero bias compared with buoy skin SSTs and standard deviations less than 0.5 Kelvin under daytime and nighttime conditions [source]

“Skin” temperature is “a few millimaters thick”…..(3mm is about 1/10th of an inch) and tells us almost nothing about energy content of the sea water below it.

It is much like forehead skin temperature of children — a quick and easy measure ($9.99 at CVS) but will read fever when a child is flushed and not running a fever at all.

I’ve always found the idea of an average temperature, wither it’s for a country or continent or planet, to be absurd. In such a large system the task of coming up with a single number that represents the whole is a fools errand.

Kip, from the very Wikipedia article to which you link (which also mentions that skew is counter-intuitive):
negative skew: The left tail is longer; the mass of the distribution is concentrated on the right of the figure. The distribution is said to be left-skewed, left-tailed, or skewed to the left, despite the fact that the curve itself appears to be skewed or leaning to the right; left instead refers to the left tail being drawn out and, often, the mean being skewed to the left of a typical center of the data. A left-skewed distribution usually appears as a right-leaning curve.

Ian,
Correct another quick way to determine direction of skew is, “The Mean always chases the tail.” The median is resistant to outliers, while the mean is not. I teach it this way: “If you wan to impress someone you are interviewing to move to your company and city, you would say that the Median price of a house is….. and the Mean (average) salary is…..

While you never get too old to be refreshed, one would think that anyone doing science would not need a primer in averages, at least until you get to the statistical study of distributions and other uses like normalization. Noting a problem, about 1990 I started teaching about ‘logical errors’ to biology students, including those in ecology and environmental assessment. These have fancy latin names, most not in much usage as also their english equivalents. It is not difficult to find examples in ‘scientific papers.’ One, of course, it gets you in trouble with linear extrapolations. These should be in basic education and would have nothing to do with politics despite being in common use there. Argumentum ad misericordiam may be part of the climate and other environmental errors.

I have also watched weather forecasters since they appeared on TV, and despite their great improvements you still get average temperatures listed as ‘normal.’ Explaining this, which some do, is a lot easier than the complexities of rip currents, which they locally warn about. “Feel good” temperatures are a imprecise, but useful metric, but that language strikes me as talking down to people.

I will look for ‘fruit salad’ examples, maybe one so old to be putrefied. It would be useful to have a list and analysis of logical errors. I have a list somewhere that I used, but I think I have an error in it.

Kip,“Sets to be averaged must be homogeneous”
There is actually nothing wrong with fruit salad. It is a popular and nutritious dish. I like it. The question is, why are you averaging? It is usually to get an estimate of a population average. So it is associated with a sampling problem. That is where homogeneity counts. But inhomogeneity doesn’t mean impossible – it just requires more care to avoid bias.

Polling is a case in point. It does predict elections, mostly. Averaging is useful, and people pay for it to be done well. But the difference between a good and bad polling is in sampling, because the population is heterogeneous. Men and women, say, think differently; that doesn’t invalidate polls, but it means you have to take care to get them represented in the right proportion.

The reason for care is that you easily might have a biased sample because of method. Blood groups are heterogeneous too, but pollsters don’t worry about that, because they have no evidence that it affects opinion. But even if it did, they can’t practically determine it in a polling sample, and it is unlikely that polling methods would systematically favour A over B, say. It would be part of the general randomness.

Homogeneity and sampling error is the basic reason for using anomalies in temperature averaging. Temperatures themselves are inhomogeneous, as you describe in the lake example. And it is very hard to get the sampling proportions right – in fact, it isn’t even clear what “right” is. But when you subtract the mean for each site, the anomaly is much more homogeneous, and good sampling is much less critical. That is just as well, because the ability to choose samples is limited. I write a lot about this on my blog – I set out the sum of squares arithmetic here.

Nick ==> I like fruit salad too, preferably with mini-marshmallows. And while not the apex of nutrition, it has its place, just like some of the Fruit Salad metrics being [mis-]used in Climate Science today.

“… in fact, it isn’t even clear what “right” is.” You hit the nail on the head here — when trying to “average” heterogeneous objects, there is no logically and scientifically “right”. In fact, no matter how one does it, there is no guarantee that the result (right or wrong) actually means anything useful. I don’t object to the methods — it is the failure of logic and the assignment of dubious meaning to the results.

“But when you subtract the mean for each site, the anomaly is much more homogeneous, and good sampling is much less critical.” I am afraid it is my opinion, with the above paragraph in mind, that when you subtract the means, you end up with the Fruit Salad’s mayonnaise, which while an interesting new substance (fruity mayo) it is not particularly enlightening. [It may have a use for something — but it ain’t nutrition.]

Nick, in a nutshell your reply illustrates why you are almost always wrong about everything. You go off the rails when “It is usually to get an estimate of a population average.” From there it is impossible to recover.

I would explain but you would not understand. Oh, and give up on the anomalies crap.

Stokes,
You said, “The question is, why are you averaging? It is usually to get an estimate of a population average.” Why are you trying “to get an estimate of a population average?” Your explanation is circular and doesn’t really explain anything.

“Your explanation is circular”
No, it is standard stats. The arithmetic done is averaging a sample. That is to serve as an estimate of a population mean. That means that you have to define the population (which is said to be inadequately done here), and establish that the sample is representative (or re-weight if it isn’t).

Nick ==> Let me repeat: “You hit the nail on the head here — when trying to “average” heterogeneous objects, there is no logically and scientifically “right”. In fact, no matter how one does it, there is no guarantee that the result (right or wrong) actually means anything useful. I don’t object to the methods — it is the failure of logic and the assignment of dubious meaning to the results.”

“when trying to “average” heterogeneous objects, there is no logically and scientifically “right”. “
We’re always averaging heterogeneous objects. If they were totally homogeneous, there would be analysis needed. Take the Dow – much quoted by all sorts of authoritative people. It’s an average of averages of all sorts of entities. There is a lot that it doesn’t tell you about details. But few ignore it.

Or take your hypertension example. What is hypertension? It is BP considerably above average. Again, not to be ignored.

Nick ==> You mustn’t confuse and conflate the objects with the data. We don’t average weight and height (we might for some purpose find a ratio and an average – a pdf of a ratio) but weight and height are incommensurable and can not be averaged. It is senseless to find an average between the largest and smallest animal in Africa, the vast difference in size between elephants and microbes makes it a logically silly exercise, they are logically too heterogeneous.
The Dow Jones Industrial Average is a weighted mean of the prices of 25 industrial stocks, different stocks to be sure, but the average is of prices (believed to be a measure of investor confidence), weighted to allow changes in individual stocks to affect the Dow evenly.
The HBP example uses a species wide average to find a non-disease-state range for the metric. A correct and useful example of averaging. In many cases in medicine, each patient has a ‘normal’ range of various body metrics which their doctor’s look to to inform them if something is beginning to go amiss.
Averages have some correct uses — that’s why we use them. averages have to be physically and logically sensible before they are calculated, with a correctly understood meaning — what the metric is going to inform us of.

They report not just median family income, but also average (mean) family income, and median and mean household income. In addition they report on per capita income, and quartile distributions for both families and households.

The point being, that all these different averages, for these various sets, are all useful.

I’ve always thought that knowing the average temperature was far less interesting than knowing the extremes. If it gets too hot my tomatoes will fail, but it’s the temperature extremes that will do them in, not the averages.

A temperature for a lake surface is probably not the result of a single measurement. It is probably an average of multiple measurements. I work with a physicist who keeps me on the straight and narrow of these matters and he says it is legitimate to average a set of numbers that are themselves averages, only on condition that the results are expected to be the same. So I can take 30 measurements about the lake and average them to get a average temperature for that lake’s surface. Yes or no? Do I expect them to be the same?

If the temperature ‘is expected to be the same’ I can use that average as a if it is a single accurate representation of the lake’s surface, correct?

If the paper claimed to have found an increase or decrease in the average of a number of lakes’ average temperatures, their claim is not very useful because the question is not one that leads to useful information.

However, just looking at the averaging of averages, when is enough averaging too much? In high school we were taught that you cannot average a pair of ratios. Well, averages are ratios. An average of a bunch of ratios each of which is an average of a bunch of other ratios seems to violate a pretty basic rule of mathematics.

If I take a number of measurements of a single lake, do I really expect the numbers to be the same? Does each measurement represent something inherent about the lake? In the shallows the temps will be higher, in the deeper areas, lower. I do not really expect the numbers to be the same. Now I average the numbers anyway, learning nearly nothing about the lake’s heat content, and move on to averaging the final number with those from other lakes.

Do I expect them to be the same? No. They are different lakes. Do I expect the average to represent the lake as accurately (or inaccurately) as the first lake? Nope. It will have a different shallow-vs-deep ratio fundamentally compromising the set’s ‘sameness’.

The average altitude may be different. It seems to be there are too many averages included in the analysis. Is that true?

If the atmosphere were warmer and the relative humidity lower, evaporation from the lakes would increase, lowering the surface temperature. Lake surface cooling could be an indicator of a warming world. I think their calculated number does not contain any information. It is just a number. You could divide it by Tuesday and multiply by purple and get another number. So what? Fruit salad.

Crispin ==> You might want to take a pass at trying an essay on what is procedurally wrong with the lakes study — there is certainly enough there to keep you busy…you seem to grasp some of the problems. links are in the essay and the SM is available.

Crispin“he says it is legitimate to average a set of numbers that are themselves averages, only on condition that the results are expected to be the same”
That is the homogeneity issue. It isn’t an absolute condition; it’s just that if you don’t have it, more care is required. To go back to the polling example, you probably don’t expect the same answer asking women as for men, on average. So you make sure you include the right proportions.

“In the shallows the temps will be higher, in the deeper areas, lower. I do not really expect the numbers to be the same.”
They deal with that:“For each lake, we calculated the temperature anomalies relative to its 1985–2009 mean.”
That is the point of anomalies – you take out the part that you expect to be different (deep, shallow etc). The anomalies you do expect to be the same, from the way you created them. Or rather, the expected value of the differences is zero. From an inhomogeneous set, you create something much more homogeneous.

Crispin, Nick ==> I will, later in this series, deal with the train-wreck of problems caused by averaging averages of averages.
The problem is that in the Lakes example, they did not have any valid averages to begin with — they have just grabbed numbers that happen to be available (the data set of satellite estimated skin surface readings from MODIS — one number for each time period). There has not been any real effort to create actually correct water temperature data sets (except in those lakes being sampled by hand or buoy). Satellites do not sample water or temperatures. They end up with a mechanical average of a non-physical property “skin surface temperature” — in this case “summertime nighttime skin surface temperature”.
Look at the video from the AGU meeting and see the image of Lake Tahoe on a typical, windy day. There is no real average, the lakes data set is not made up of real averages, the anomalies of not-real averages of not-real averages are not scientifically valid for purposes of determining 1 degree difference over time.

I work in the horrible world of rating performance, especially efficiencies, which are ratios. People will happily operate a device under 4 different loads, calculate the energy efficiency for each condition, then simple-average the four numbers and claim to have given it an average efficiency” rating. It is just number-mumbling.

Even if there was a duty cycle that involved those particular loadings for that particular duration each, each individual calculation hides so much information the the final ratio of ratios tells us little about how it will perform.

WRT the lakes, it is obvious that the intent is to communicate that the entire lake temperature has changed, the enthalpy has risen or dropped, using the surface temperature as a proxy. It is nearly useless as a proxy for communicating, ‘Warming is upon us!’ Maybe I can go so far as to say it is literally useless.

If you want to estimate the heat content of a lake, you could do it with a reasonable number of measurements, and set a target error band. Suppose you accepted to have a number with an uncertainty of 50%. Let’s say the heat content is 1 TJ, ±0.5 TJ. Anyone reading this think they could get that from a satellite surface temperature reading?

Nick suggests it is all about anomalies. I understand the concept so no need to repeat. To get two readings a year apart and then claim that the bulk temperature of the lake was known to within 50%, and then claim that the anomaly was meaningful would require one helluva a difference in surface temperature. I am not sure if most of the readers will know immediately why, so give me one more paragraph before the numbers flow.

If the anomaly is smaller than the uncertainty of the total energy content, the anomaly tells us exactly nothing. If the surface temperature one year is 10 and the next year it is 11, and the uncertainty about the total energy stored in the lake is 15%, one cannot claim to have detected a difference so the proxy fails.

How good would the total energy number have to be in order for a 1 deg anomaly to mean anything real?
10 C = 283 K
11 C = 284 K
The heat content increase (assuming the lake contains exactly the same amount of water both years) is 0.35%. How many readings would one have to make in a lake to get the heat content to within 0.35%? That is 1 TJ ±0.0035 TJ. One part in two hundred and eighty-three. One?!?

Similarly, a measured increase of 0.1 degrees would have to be backed up with a way of modeling the total energy to another significant digit. 1 TJ ±0.00035 TJ. Good luck with that. If the limit of determination is 200 times the anomaly, you can claim absolutely nothing. Think about an ‘average global temperature’ increase of +0.001˚ ±0.2.

It is clownish efforts like this which render ‘ocean acidification’ numbers meaningless. The pH of the ocean can change 1 unit in 2 metres. Collect a trillion well-distributed one-off pH measurements. What is the uncertainty? Do it again after ten years. What’s the change and the uncertainty? Can’t claim a thing.

They make no attempt at heat content or bulk temperature at all — their summertime nighttime skin surface water temperatures are mechanically estimated by the method of detection (satellite MODIS). How in the world they arrive at an single-number average surface water temperature for both Lake Tahoe (which has huge spacial temperatures differences under normal circumstances) and Lake Superior (an inland freshwater sea) is not explained anywhere in the study or SMs.

“The heat content increase (assuming the lake contains exactly the same amount of water both years) is 0.35%.”
This is nonsense. On this basis, you could never know the temperature of anything. You couldn’t measure if your personal heat content increased by 1%, right? It couldn’t have any effect, could it?

Lakes temperature vary with water content in the lake at the time of measurements. Water in the lake vary with rainfall and silting and as well encroachments. Thus, averaging of lakes temperature, really has no meaning like our global average temperature anomaly. They should be studied individually and characterise the changes in temperature of that lake, will have some meaning.

Hyderabad [where I live] get water from two reservoirs [Himayatsagar and Osmansagar]. The rainfall shows a year to year variation with little or no trend but the inflows show a gradually decrease. Lakes are now getting wastewater and thus changing the chemistry of lake water and thus temperature — this is not associated with climate but human greed.

An aside Dr. S.J; I worked many years in water resources and sewage treatment. In several cases, adding properly treated effluent to receiving waters actually IMPROVED the water quality and stabilized the water levels. Waste water used for agriculture and street boulevard irrigation are also a good applications that have been used for decades. Yes, removing water from rivers for irrigation can be a big problem. Hopefully we can learn a bit as we go along. The other side of the argument your comment reminds me of, is the claim that oceans are rising because of our “mining” of ground water resources. I doubt that the ocean is rising from this source as we also have some folks claiming that increased rainfall is depleting the water in the ocean. Seems each issue can be argued several ways.

The point of my comment is that we can be hopeful for the future in spite of the alarmists out there. There will always be alarmists about something.

Pessimist: The glass is half empty.
Optimist: The glass is half full.
Engineer: Use a smaller glass.

I live in Hyderabad city. Here sewage of 2000 mld is generated [during rainy season, rainwater is added to this]. Government established STPs to treat around 700 mld but in reality treatment is carried over of less than half. The untreated and treated sewage along with untreated-partially treated industrial effluents join the River Musi. Using this water food-milk-meat is produced and supplied to city. The lakes in city are cesspools of poison as sewage is directly released in to them.. I am one of the few environmental groups fighting against this menace with no success. The success depends upon the governance.

Another interesting and thoughtful essay. I’m fine with your major points. Some rather scattershot observations, none of which really affect your arguments/conclusions much.

1. There’s what looks to be a good article on lake surface temperatures here. http://faculty.gvsu.edu/videticp/stratification.htm If I understand it, the “nightime skin SST temperature” is the temperature of the epilimnion — the well oxygenated surface layer which is well mixed by winds (and in windless periods …?) Anyway, I’ll just point you to the article.

2. There are a number of different means computed in different ways, some of which are actually useful. The one we are all familiar with is the arithmetic mean. But sometimes the geometric mean or the harmonic mean are more appropriate. There are probably yet other means that might be appropriate in some cases.

3. The hypertension chart is certainly an example of a messy data set, and it may well be an example of a collection of data that is not meaningfully analyzable with the usual tools. It may also be an example of how not to handle data if you want useful results.

Blood pressure is easily measured with non-intrusive instruments (and also easily mis-measured, but that’s another story). As a result it has been the subject of tens of thousands of published papers. I’ve read a fair sampling of the papers over the years, and have concluded that for the most part those folks don’t seem to have the slightest idea what they are about. They seem for the most part to be treating something that is often a symptom as if it were a disease. Where it is recognized as a symptom, I don’t have any problem with treating underlying causes of hypertension like kidney disease. That’s a great idea where it works. But most treatment appears to consist of cosmetically using drugs to reduce blood pressure with no idea of whether reduction is necessary, if/why reduction has any useful effect, or even if reduction is a good idea.

4. I don’t have any argument with the notion that planetary temperature is a badly defined and non-rigorous metric. If nothing else, it’s far too dependent on the state of ENSO to be treated with the respect it is given politically. But we probably do need a simple metric for the state of planetary glaciation and it’s not that easy to see a better alternative.

You can average averages in some cases using the harmonic mean. A great example for teaching is ‘average miles per gallon’ v.s. ‘average litres per hundred kilometers’. They have to be calculated differently.

Afterthought: Reading randomly through the excellent commentary, it seems to me that much of the discussion is about sampling theory rather than averages per se. Ideally a sample is representative of the whole and large enough not to be skewed too badly by outliers. That’s simple in theory, but a nightmare in actuality.

Another afterthought. What we want to know is rarely something we can measure directly. So we measure what we can and hope it is close enough to what we want to know to produce a useful result. e.g. We measure the night time surface skin temperature of a spot somewhere on a lake and hope it is fairly representative of the temperatures of the lake as a whole. (And, BTW, what one gets from MODIS looks to be be a set of “mechanical” averages over a region somewhat larger than 1km diameter at slightly different times and locations on different days).

Are you planning to address sampling theory and the delta between what we can measure and what we want to know in future essays.?

Don K ==> Yes, one of the problems of the study, if I were to critique it (which I am not doing here), is the fact they average false-averages (the initial measurement of skin surface temperature from
“spot somewhere on a lake”, some randomly chosen 1 km grid cell, presented as the average surface temperature for that data point), which is equivalent to a mechanical average (determined by the equipment used, with no logic or intelligence involved). These false-daily/nightly averages are then averaged to a monthly/yearly average then those to a global average (or maybe different steps, don’t remember)…but in any case, they are certainly committing the offense of averaging averages of averages (which I will hopefully discuss in a future part of this series.)
Simply saying that because our method is consistent — even though we know it does not produce the actual results we claim, even though it produces false-average surface temperatures for the lakes at each measurement — we can nonetheless claim that our long-term trend is valid — is Madness!

Very nice essay, Mr. Hansen. You did omit one additional definition of “average” that I always present to my engineering students, and which is pertinent to the concept of average Earth temperature–the midpoint of the maximum and minimum.

Also the phrase ” In order calculate” in your summary paragraphs ought to be ” In order to calculate”

The mere fact that it exists does not mean that it is knowable. It is true that the “real” temperature has precise physical meaning, but it is not true that the average calculated from the data we have has even marginal physical meaning.

Averaging intensive variables is always a problem when you don’t know how much stuff there is. The lake example illustrates this problem quite well. It is much easier to see when you have to very disproportionate areas (volumes) that you are averaging. Area A1 has temperature T1, and area A2 has temperature T2, but area A1 is much larger than area A2. The “average” temperature is much closer to T1 than T2 (assuming they are different values as well). The same is true for averaging temperatures within the atmosphere.

Now we add in the problem of incomplete sampling and the precision becomes even more ambiguous. How can we know what the temperature is when we only cover a portion of the planet with sensors (a portion that does not properly meet sampling theorem limits)?

Then we have to consider the truth that our atmosphere’s temperature, whatever it is, changes constantly. What does an average mean when it is fluctuating non-randomly (and randomly) over time?

In general, averages don’t always (or often) have physical meaning. Rarely does the mean and meaning exist in a set itself (only the mode is guaranteed to be a member of the set). That there is a temperature and it has a precise physical meaning is meaningless, so claiming this as some sort of “truth” is silly. It allows us to make general statements like “it is warmer on the planet earth than it is on Pluto” and not much more.

I commented above about definitions. To follow up on that and respond to your comment, if we are trying to determine the average temperature of Earth, and we know that we don’t have complete or proper sampling, then we cannot compute what we are looking for. At best, we can claim to have computed an average temperature of the available data set, which almost certainly isn’t the average we are looking for.

The mode may not even be a member of a set because it may depend on how we bin the data. Also, there may be a tie for the most frequent value(s).

The mean of min and max temperatures is that of the probe that is in completely different environments for each measure. When I became interested in this a few years ago, I found it strange that people were studying the trend in the global mean of these. I found it a complete joke that its pretty much the mean of infilled data obtained from actual measurements.
Surely as an indicator of very small change, you would look at the trends in the data that you have, only. If you can’t get something meaningful from that, you are not getting anything from a trend in global average if infilled data.

I have dealt with these issues since I taught my first math class to young lads and lasses who where more interested in getting laid than in statistics. (although, statistics could be useful to them even in that endeavor one would think)

I can not even wrap my head around “averaging” temperatures on this planet since it is the energy content that matters. Water, rock, and air at the same temperature would have different energy content, or so my science teachers said back in the early 70s. Do we still admit that even with climate alarmism being the driving force behind government science today?

Mark,
If the ocean and air temperatures were reported separately, then they would serve as proxies for the energy in the two domains. However, with something like a 4-fold difference in the heat capacity of water and other materials, averaging the temperatures together is comparing avocadoes with fruit salad.

They would be somewhat better proxies – but, in my opinion, still not very good ones. The energy content of a cubic meter of air at 70% saturation is very different from that of a cubic meter of air at 7% saturation, even if they have the same temperature. Unless you are very careful in your selection, a cubic meter of seawater in one place has a different chemical composition than a cubic meter in some other place – with a corresponding difference in energy content even if at the same temperature.

Separating water and air only removes the biggest energy content differences – and we are commonly talking about rather small energy differences when saying something has “warmed” or “cooled” by tiny fractions of a degree.

About 1% change in salinity makes a 0.1% change in specific heat capacity. Not a lot but 0.1% of 300k is 0.3 degrees. How much has the average ocean temp changed? Something like 1/100 the of that in the Argo data?

The problem I have with averaging is that it eliminates all of the specific information that might have been useful in projecting possible future scenarios for any given area. Climate by definition is regional. It consists of a range of weather patterns and temperatures common to a particular geographic area. Once temperatures all over the globe have been averaged together, it is then impossible to tease out any useful information about what that might mean for the climate, or even weather over a particular region in the future. It would be like averaging together all of the baseball scores over the 20th century and comparing it to the ongoing 21st century. If the average goes up, or down, or stays the same…what does that mean for next week’s score at the local stadium?
A global average is not useful information, especially since the area being measured isn’t the whole globe, but just an arbitrary 2 meters above the surface, except when temperatures measured outside that sliver of the world, deep in the oceans, or under the arctic ice, might help support someone’s contention.
Climate must be studied regionally to remain useful. Averaging seems to primarily be used to con people into thinking that if something scary is happening somewhere, it must be happening a little bit, close to where you live.

Agreed Hoyt. The next problem is that having thrown away almost all useful information by taking anomalies and averaging, alarmists are then doomed to be unable to explain what is happening at the local level.

My view is that the proper starting point is local with examination of any patterns observed. So, as I frequently ask, what was the climate 30 years ago where I live and what is it now.

“The global temperature exists. It has a precise physical meaning” No, it does not. For very simple systems it can be shown to be nonsense. Cooling systems would show pseudo-warming, warming systems would show pseudo-cooling (that is, ‘cooling’ based on the global pseudo-temperature nonsense). As for ‘warmer’ and ‘cooler’ that works only for largely non-overlapping range of temperatures. If they overlap a lot, it becomes nonsense.

Agreed, but the statement you quote is just mindless rhetoric from Nick Stokes. I can’t quite figure out why the author included that statement in his article.

It’s the same thing with Mossshhher the Great and Powerful. He should be along shortly to demand that the world explains how his data fiddling could possibly not be revealing the word of God. Or something.

Hopefully this article will educate. But then again it is hard to convince somebody he is wrong when his pay packet depends on his being wrong.

Surprisingly, the averaging that is most often used with climate data, i.e., the mid-range value produced by the mean of daily Tmax and Tmin, is nowhere mentioned in this far-ranging tract. That confounding average, along with the mixed-type, heterogeneous temperatures (1.5-meter dry-bulb, SST) used in manufacturing “global” indices, is what renders the fruit salad so scientifically sour. And that bad taste is only multiplied by various ad hoc adjustments that provide the salad dressing.

Todays recorded temperature that will become part of the historical record will begin at 4 pm my local time starting in Greenwich and the last temperature of “today” will be recorded well after my local day has ended (midnight). So even the average global temperature for a day doesn’t include a real day.

Taking a human patient’s temperature is analogous. It varies from person to person, there are all sorts of errors, it tells us nothing about fingers and toes, but it is still pretty accurate and quite useful. If it starts ‘trending’ – there may be a problem.

My opinion is that, likewise, global averages are pretty accurate and quite useful if they reveal a trend.

The problem many here have (imo) is not so much with how it is determined but that it appears to be trending in a certain direction. This post would not exist if they were going down.

Tony,
Taking a person’s temperature is only analogous to local sampling to determine what is going on locally. You would never take the temperature of 1000 random people and use the average to determine the health of the whole human population. Calling a global average “accurate” doesn’t make a lot of sense because a global average isn’t an actual measurable thing. It is a calculation. If it trends up, something might be happening. If it trends down, something might be happening. But if it stays the same, guess what, something might be happening. If the Earth began to exhibit wider extremes in temperature, say 20 degrees hotter in the day and 20 degrees colder at night, that would be something big, but it might average out to exactly the same average temperature we have today. On the other hand, if the Earth became less extreme, and temps were 30 degrees less warm in the day, and 30 degrees warmer at night, It could still theoretically average out the same, or with a trend up or down. That’s how averages hide what might really be going on.
If I wanted to, I could sample the light coming from a rainbow and register the wavelengths of all of the different colors of light and average them together. I would end up with an average wavelength that would correspond to a specific color, but calling that average color accurate is nonsense because calculating the “average color” of a rainbow is completely meaningless. Think of the Earth as a rainbow of temperatures. Averaging them all together is not data, and has no meaning.
In simpler terms, suppose I offered you one of two cups of coffee and you asked me if it’s hot. I might say that the average temperature of the two cups is 100f. Which cup do you choose and why?

It is analogoes to one person in that each individual is made up of countless sytems and cells and that a temperature meaurement of say 37.6 is just an otherwise meaningless number. If it begins to change – that is indicative and thus meaningful, as is the rate of change.

“On the other hand, if the Earth became less extreme, and temps were 30 degrees less warm in the day, and 30 degrees warmer at night, It could still theoretically average out the same, or with a trend up or down. That’s how averages hide what might really be going on.”

Perfectly true, but if the average is moving it incicates something probably IS going on.

tony mcleod ==> A changes in temperature, which you call a trend, only tell you that the trend (by whatever method you used to measure, compute and determine trend) exists for the period of time you have selected. It tells you literally nothing else. Quite literally, we have simply no idea why temperatures go up and down on yearly or decadal scales — just some associations with climate phenomena – like ENSO, AMO, etc.

“A changes in temperature, which you call a trend, only tell you that the trend (by whatever method you used to measure, compute and determine trend) exists for the period of time you have selected.”

Hmm. A trends indicate a trend.

There is an upward trend. In conjunction with other data like sea-ice area –

-a sensitive yet slow-moving indictor of wider area temperatures, it may be indicative of a warming ocean/atmosphere.
(On a side-note: maximum area moving from early November to mid-June. A new, geo-engineered dipole?)

I don’t think it average temperature needs to be dicarded.

“we have simply no idea”
What should we be measuring to find out? Do you think it matters?

Like all cultists of your ilk, you fail to realize that trends of ten, 20 or 39 years mean nothing in terms of climate, which is naturally cyclic.

There is zero evidence supporting the repeatedly falsified, baseless assertion that humans are responsible for whatever warming might actually have occurred between 1977 and late in the last or early in this century. And all the evidence in the world that mankind had nothing whatsoever to do with it, as it was well within normal bounds.

Let alone whether slight warming, from whatever cause, is good or bad. So far more plant food in the air has been greatly beneficial.

And please explain why for most of the postwar period, during which CO2 has steadily risen, global temperature has either fallen dramatically or stayed about even. Thanks.

All measurements are accompanied by an uncertainty. That is a physical fact. The true vale of any measurement is not knowable. When the difference between two measurements is smaller than the uncertainty in each of them, no claim for ‘difference’ can be sustained.

It gets worse. If the difference is not three times the limit of detection, by convention it is not accepted as being different. Consider the ocean temperature-based claims for heat content. The measurements are made to 0.01 degrees C. The uncertainty is 0.02 degrees. The claim that the oceans were 0.001 degrees warmer than ‘last time’ is unsustainable, based on the limit of determination and the uncertainty.

Now, check the measurements for the average temperature of the world that underpin the claims that 2015 was hotter than 2014 and 2016 was hotter than 2015. The claimed increase is an order of magnitude less than the uncertainty. And on this we should spend 100 trillion Dollars, just in case.

Perhaps you could amend you question to include the uncertainty. I know the ordinary man-in-the-street does not think this way, but experimentalists do.

You could say hmm, its rising at .1C ±0.5/decade for a hundred years, wuwt rate of change?

That gives 1.0 ±0.5 degrees per century. So what? Since when has warmer temperatures over the whole globe, accentuated in the Arctic, been a cause of loss and misery? It is easy to show that the opposite, cooling of 1C, has been.

8000 years ago the world was about 3 degrees C warmer than now. The Sahara and Gobi deserts were grasslands. The Arctic was largely ice-free in summer. That’s good medicine. Swallow it. It’s good for you.

If we are in for 3 centuries of warming at 1 C per century, it will be fabulous. I see no hope of that at present because we are going into a pretty steep decline, but hopefully the sun will wake up in a few decades. In the medium term, 5000 years, we are fatefully headed into the next ice age.

My opinion is that, likewise, global averages are pretty accurate and quite useful if they reveal a trend.

This blind article of faith about GAST anomalies is wrong on two levels:

1. Accuracy is nowhere scientifically tested, let alone validated, in compiling global indices. What is mistaken for accuracy is the reproducibility of results from effectively the same unevenly-distributed data base of deeply flawed and variously biased records. This leaves unanswered the critical question of how representative is that data base of the actual field of temperatures on the globe at any given time.

2. Trends are useful only if they are truly secular and pertain to the same location. Given the presence of strong oscillatory components in many local variations of temperature that reach well into the centennial range, shorter-term “trends” are highly changeable and offer no predictive value. Moreover, none of the indices that purport to represent GAST keep the measurement locations fixed throughout the entire duration of the index. That’s analogous to diagnosing a developing fever by comparing the temperatures of different patients.

The popular indices (which have steadily minimized the effects of multidecadal oscillations and increased the steepness of “trends”) have been elevated to the status of iconic measures precisely to ingrain such blind faith in the scientifically unaware.

“Lakes (at least Summertime Nighttime Lake Skin Surface Temperatures) may be warming, but they are not warming even in step with air temperatures, not reliably in step with any other particular geomorphic or climatic factor….”

S/B: nor reliably in step

“…Moreover, Fruit Salad averages not only can lead us astray on a topic but they obscure more information that they illuminate…”

The physical meaning is very precise. You need to review operationalism. And your fruit salad analogy is bad science and worse philosophy. Please stop that.
The facts are pretty clear.
1 c02 is a ghg.
2. Humans are adding c02 to the atmosphere.
3. GHGS warm the planet.
Notwithstanding the difficultie’s and nuances and technical details we have known this since 1896.
The earth is not flat.
We did land on the moon.

1. CO2 is a radiative gas. that is used in greenhouses because it promotes plant growth
2. Yes.. Thank goodness, CO2 has been dangerously low for many 100, or 1000s of years. Still more needed.
3. Prove it. !! By actual measurement… you know “science”, not, “climate science”

As usual, you are trivializing a complex system to attempt to appear to be correct.

1 c02 is a ghg.
CO2 behaves as a GHG in a controlled experiment where all other variables are held constant. However, in the real world, all variables are free to change and interact. Therein lies the rub! It is like making the observation that fission of unstable isotopes can lead to a runaway chain reaction and once it starts nothing can stop it. That is ignoring the use of neutron moderators to control the reaction rate in a reactor. The climate system almost certainly has natural negative feedback loops to control the effect of CO2.

2. Humans are adding c02 to the atmosphere.
Yes, but unless it can be demonstrated that in the real world the anthropogenic CO2 has a significant impact, the statement is a non sequitur. You are attempting to assign blame to humans for something that may be trivial in its impact.

3. GHGS warm the planet.
Probably, but what is in dispute is the amount that CO2 is warming Earth. It is a complex, non-linear feedback system that is poorly characterized, and there are some who say that weather and climate are chaotic systems [e.g. IPCC] that cannot be predicted with skill and therefore reliable warming estimates cannot be assigned to anthropogenic CO2. Even allowing for a positive effect, the sensitivity of the climate to CO2 is probably much smaller than generally claimed, as evidenced by the lack of the Earth experiencing a ‘Tipping Point’ in the geologic past when CO2 levels were at least an order of magnitude larger than they are expected to be when we run out of abundant fossil fuels.

It has been my experience that people who are self-educated, or have learned on the job, usually have gaps in their knowledge that they don’t even realize. Thus, they are all too quick to assume an outcome based on their limited knowledge. Certainty is inversely proportional to expertise. That is, the more expert someone truly is, the more likely they are to have reservations about predictions. They are aware of all the things that haven’t been taken into consideration.

Clyde==> “It has been my experience that people who are self-educated, or have learned on the job, usually have gaps in their knowledge that they don’t even realize. Thus, they are all too quick to assume an outcome based on their limited knowledge. Certainty is inversely proportional to expertise. That is, the more expert someone truly is, the more likely they are to have reservations about predictions. They are aware of all the things that haven’t been taken into consideration”.

Thank you!!! This is the exact situation I deal with every single day in my job. As I’m the only non self-taught person in my office, I deal with this so much I fear I am starting to go insane. The certainty with which my “colleagues” spout utter nonsense in lieu of technical explanations often leaves me speechless, and the fact that I’m the only one who realizes this fact leaves me depressed. So finding out that someone else recognizes this simple truth is a minor miracle. I will frame your words above my office desk. Yet another reason WUWT is worth every minute I spend reading it. Thanks again!

Andre,
As I get older, I find that my memory is not as good as it once was. I’m not as quick as I once was. I often struggle to find the right word. However, like a good whiskey that gets better with age, I synthesize all of my important life-observations into a coherent world view that I’m pleased to discover that others recognize as wisdom.

The facts are pretty clear.
1 CO2 is the second most important GHG.
2. Humans are adding H2O and CO2 to the atmosphere.
3. GHGS warm the planet.
4. Any increased H2O condenses and shades the planet
5. The cooling provided by 1 degree of warming and the consequent condensation of H2O is 20 times greater than the warming effect of doubling CO2.

>Notwithstanding the difficultie’s and nuances and technical details we have known this since 1896.

One of the ‘technical details’ to consider is the withdrawal of that paper and the correction published after years of better and wiser experts pointing out fundamental flaws in the theory and his claimed magnitude of warming. And that was before there was any common understanding of the cooling effect of an increase in tropical sea evaporation with increased sea temperatures.

The earth is not flat.
We did land on the moon.
CO2 has little net effect on the temperature of the planet because it is not, by far, the most influential GHG.
Svensmark is correct and deserves the Nobel Prize for Physics.

“One of the ‘technical details’ to consider is the withdrawal of that paper and the correction published”

Making up stuff. There was no withdrawal and no correction. It is here. One of the Angstroms disagreed, but Arrhenius rejected that, and turned out to be right. In 1908here“”If the quantity of carbonic acid [ CO2 + H2O → H2CO3 (carbonic acid) ] in the air should sink to one-half its present percentage, the temperature would fall by about 4°; a diminution to one-quarter would reduce the temperature by 8°. On the other hand, any doubling of the percentage of carbon dioxide in the air would raise the temperature of the earth’s surface by 4°; and if the carbon dioxide were increased fourfold, the temperature would rise by 8°.””

1908? Cherry picking!
Check the 1915 papers.
He did give up the defense of his original error.
Doubling the % of CO2 produces nothing like a 4 degrees of warming. But you already knew that.

Arrhenius discounted his original 1895 (not 1896) by more than 50% after being convinced the critics were correct. Yes, he battled gamely, but right from the first publication, his bad math was attacked. In the end he relented.

I said ghgs can’t warm air, some can: but they most certainly can’t warm the atmosphere overall. They constitute a refractive insulating layer between the rock of the earth and the fire illuminating, and warming it.

Kip,
I think you are doing the authors of the papers about the lake temperature a real dis-service using them as an example for the mis-use of averages. Certainly they state the average result in their abstract but then spend the rest of the paper explaining the whole distribution of temperatures changes. They also show the whole distribution of temperatures in their paper. And while for an arbitrary distribution the average does not tell the whole story it is does provide the most information of any moment used to describe the distribution. Hence since space is limited in the abstract of a paper I would expect the authors to state the average and then in the body explain the rest of the distribution.

Germinio: there’s considerable truth in what you say. I’ve not read the underlying paper, but Kip’s analysis indicates that the authors use their puree of fruit metric to establish a trend. When the studied population lacks homogeneity, trend analysis can go wrong.

Germinio ==> This essay is not a critique of the paper — I use their paper as an example of the misuse of averages of incommensurable, heterogeneous measurements — Fruit Salad averages.
The paper abstract contains the offense. The Press Release is far worse, as is the AGU presentation. You must actually look at all of these, and the video, to see that their offense is in the USE is an invalid average — a Fruit Salad metric – one that is “physically and logically flawed beyond any use [except for propaganda purposes]. ”

They get credit for the “more scientifically valid findings of the study which show that each lake’s temperature is changing due to local, sometimes even individual, geomorphic and climate causes specific to each lake and casting doubt on the idea of global or regional causes.”

If you’d like to write a full review/critique of the Lakes study, I’d be happy to see it here….but you must include all four elements: the study as published, with all SMs; the AGU Press Release; the AGU video; and the lakes project poster.

Your opinion may very from mine –

but overall, they seem to me to obviously create a scientifically invalid, fruit-salad, average of heterogeneous data to declare global trend purely for propaganda purposes.

Kip,
I would disagree that the paper mis-uses averages. It uses averages as one part of a set of tools all of which
provide some information. Also more fundamentally your claims about “fruit-salad” averages does not have any precise definition. Take for example your average of the height of all animals in North America. You claim this is flawed because it gives a result that seems wrong to you. However it does highlight an crucial and often forgotten point — most of the earth’s biomass is in the form of small insects and bacteria. Focussing on large and statistically rare animals like bears (or even humans) will lead to bad and incorrect decisions about how best to look after the environment. Thus there is merit in calculating the average height of all animals.

Similarly with regards the temperature trends of lakes, taking the average and width of the distribution would tell you whether or not lakes are in some way homogenous. And looking at the distribution in the paper a surprise might well be how similar the trends are despite initially thinking the lakes are very different.

The question is not whether taking an average is valid but rather how much information about the distribution it gives. It will almost always give you more information about the distribution than any other single number. But it will not tell the whole story and the authors of the lake study do not pretend that it does.

Geronimo ==> Well, if you haven’t seen my point by now, there is no sense continuing.

To me, they purposefully average data about objects that are heterogeneous — which is their own finding — solely for the purpose of making a specific point, not for any scientific purpose but soley as propaganda. This is obvious from the four items linked: the paper, the AGU Poster, the AGU video, and the AGU Press Release.

And there is the “average” surface temperature of 15C & 288K compared to the ToA 255K at 240 W/m^2 difference of 33C which is explained only by RGHE theory.

To be 33C or not to be 33C
There is a popular fantasy that the earth is 33C warmer with an atmosphere than without due to the radiative greenhouse effect, RGHE and 0.04% atmospheric CO2.
Let’s start at the very beginning, a very good place to start – or so I hear.
The 33C difference is between an alleged average surface temperature of 288K/15C and 255K/-18C, the alleged surface temperature without an atmosphere. Let’s take a closer look.
Just which average surface temperature? The two extremes? (71C + -90C) / 2 = -10C? Or the average of all the real actual (adjusted, homogenized, corrupted) measurements 90% of which are in the US, Canada, Europe and Australia? What about the sea surface? Satellite data? Over thirty years?
Per IPCC AR5 glossary the average land surface temperature is measured 1.5 meters above the ground, but 80% of the land (Africa, Siberia, South America, SE Asia) doesn’t even have reliable weather instrumentation or data.
The average sea surface temperature is a combination of buckets and thermometers, engine cooling intakes, buoys, satellites, etc.
This composite “global” surface average temperature, one number to rule them all, must represent: both lit and dark sides, both poles, oceans, deserts, jungles and a wide range of both land and sea surfaces. The uncertainty band must be YUGE!
The 255K is a theoretical calculation using the S-B ideal BB temperature associated with the 240 W/m^2 radiative balance at the top of the – wait for it – atmosphere, i.e. 100 km.
So what would the earth be like without an atmosphere?
The average solar constant is 1,368 W/m^2 with an S-B BB temperature of 390 K or 17 C higher than the boiling point of water under sea level atmospheric pressure, which would no longer exist. The oceans would boil away removing the giga-tons of pressure that keeps the molten core in place. The molten core would push through the floor flooding the surface with dark magma changing both emissivity and albedo. With no atmosphere a steady rain of meteorites would pulverize the surface to dust same as the moon. The earth would be much like the moon with a similar albedo (0.12) and large swings in surface temperature from lit to dark sides. No clouds, no vegetation, no snow, no ice a completely different albedo, certainly not the current 30%. No molecules means no convection, conduction, latent energy and surface absorption/radiation would be anybody’s guess. Whatever the conditions of the earth would be without an atmosphere, it is most certainly NOT 240 W/m^2 and 255K.
The alleged 33C difference is between a) an average surface temperature composed of thousands of WAGs that must be +/- entire degrees and b) a theoretical temperature calculation 100 km away that cannot even be measured and c) all with an intact and fully functioning atmosphere.
The surface of the earth is warm because the atmosphere provides an insulating blanket, a thermal resistance, no different from the insulation in the ceiling and walls of a house with the temperature differential determined per the equation Q = U * A * dT, simple to verify and demonstrate. (Explains why 250 km thick atmosphere of Venus with twice the irradiance heats surface bigly compared to earth.)
A voltage difference is needed for current to flow through an electrical resistance.
A pressure difference is needed for fluid to flow through a physical resistance.
A temperature difference is needed for energy to flow, i.e. heat, through a thermal resistance.
RGHE upwelling/downwelling/”back” radiation is a fictional anti-thermodynamic non-explanation for the “33C without an atmosphere” phenomenon that doesn’t actually exist.

Nicholas you are right that the earth emits energy in different amounts at different places due to the rotation of the earth and the position relative to the sun and that it is very difficult to actually measure the individual incoming and out going energy.
Nonetheless if you step back and consider the overall picture you could possibly do all the calculations just from one site, wherever you choose.
The amount of incoming energy is well known and majority due to the sun by a very long way.
The temperature at any one spot monitored over a year will generally give a good idea of the temperature for that part of the globe at that altitude, latitude and longitude.
Energy in is known, Energy out is known, basically the same as energy in. The sun has been in place a long time so an equilibrium of sorts is a reality.
There are minor variations due to the redistribution of heat transmitting layers in the oceans and cloud cover.
The maths is pretty clear.
Earth at its size and distance from the sun can only emit the amount of energy out each 24 hours that comes in each 24 hours.
Scientists construct an overall average temp for that 24 hours.
It is a construct.
There is a height at which the (TOA) emmisions out equal those in.
That is the theoretical surface of the planet because we do not have a true, solid, no atmosphere surface.
The actual surface to us with the adjacent layer of air we measure temps in is a subset of the scientific surface.

Re albedo. From the moon fact sheet moon first earth second
Bond albedo 0.11 0.306 0.360
Geometric albedo 0.12 0.434 0.28
Black-body temperature (K) 270.4 254.0 1.065
The moon and earth are at the same distance from the sun.
Hence the surface temp no atmosphere is the same.
If earth had no atmosphere it would increase its albedo to that of the moon.

First, the speed of rotation is materially different. For example, as the speed of rotation tends towards infinity, the average surface temperature tends towards that of the warm side. Given the Earth’s rotation one would expect it to be warmer than the moon.

Second, when considering the no atmosphere concept, one should not see the planet as a barren rock, although that would be the outcome. One has to view the position as if it were still a water world, but without the claimed radiative GHE of water vapour in its atmosphere. The planet then has huge energy storage capacitors where solar irradiance is not absorbed at the surface, but rather down at depth between about 2 to 150 metres which gradually heats a huge volume of water before slowly working its way to the surface of the ocean. This has a big impact given the rotation of the planet and enables the planet to warm above that seen on the moon.

Third, even though one leaves aside the radiative GHE of water vapour in the atmosphere, one still has to consider the latent energy entrained within a humid atmosphere. A humid atmosphere gradually heats up, and then releases its energy slowly. Once again, this has a significant impact given the speed of rotation planet.

Finally, one should bear in mind that there is no measurable GHE observed on Mars, notwithstanding that the Martian atmosphere is about 96% CO2, ie., 960,000 ppm. Proponents of AGW say this is because the Martian atmosphere is very slight, it being about 1/206th that of Earth’s.

However, we are led to believe that what is warming Earth is the GHGs in Earth’s atmosphere. If that is so, then one should consider what Earth’s atmosphere would look like if all non GHGs were removed, ie., without any Nitrogen, Oxygen, Argon. When these non GHGs are removed, one is left with an atmosphere of approximately the same density as that of Mars!

In fact, notwithstanding the slight Martian atmosphere, on a molecule for molecule basis there are about 10 times as many molecules of CO2 on Mars (in its atmosphere) than there are CO2 molecules on Earth (in Earth’s atmosphere). IF CO2 is such an effective GHG, given that there are numerically 10 times as many molecules of CO2 in the Martian atmosphere compared to the number found in Earth’s atmosphere, why is no GHE observed on Mars?

First, the speed of rotation is materially different. For example, as the speed of rotation tends towards infinity, the average surface temperature tends towards that of the warm side. Given the Earth’s rotation one would expect it to be warmer than the moon.

Second, when considering the no atmosphere concept, one should not see the planet as a barren rock, although that would be the outcome. One has to view the position as if it were still a water world, but without the claimed radiative GHE of water vapour in its atmosphere. The planet then has huge energy storage capacitors where solar irradiance is not absorbed at the surface, but rather down at depth between about 2 to 150 metres which gradually heats a huge volume of water before slowly working its way to the surface of the ocean. This has a big impact given the rotation of the planet and enables the planet to warm above that seen on the moon.

Third, even though one leaves aside the radiative GHE of water vapour in the atmosphere, one still has to consider the latent energy entrained within a humid atmosphere. A humid atmosphere gradually heats up, and then releases its energy slowly. Once again, this has a significant impact given the speed of rotation planet.

Finally, one should bear in mind that there is no measurable GHE observed on Mars, notwithstanding that the Martian atmosphere is about 96% CO2, ie., 960,000 ppm. Proponents of AGW say this is because the Martian atmosphere is very slight, it being about 1/206th that of Earth’s.

However, we are led to believe that what is warming Earth is the GHGs in Earth’s atmosphere. If that is so, then one should consider what Earth’s atmosphere would look like if all non GHGs were removed, ie., without any Nitrogen, Oxygen, Argon. When these non GHGs are removed, one is left with an atmosphere of approximately the same density as that of Mars!

In fact, notwithstanding the slight Martian atmosphere, on a molecule for molecule basis there are about 10 times as many molecules of CO2 on Mars (in its atmosphere) than there are CO2 molecules on Earth (in Earth’s atmosphere). IF CO2 is such an effective GHG, given that there are numerically 10 times as many molecules of CO2 in the Martian atmosphere compared to the number found in Earth’s atmosphere, why is no GHE observed on Mars?

“First, the speed of rotation is materially different. For example, as the speed of rotation tends towards infinity, the average surface temperature tends towards that of the warm side. Given the Earth’s rotation one would expect it to be warmer than the moon.”

Intuitively that doesn’t sound right. Ratio of time absorbing incoming radiation to the time emitting is constant regardless of rotation rate. Also, the total energy absorbed must equal the total energy emitted, which results in the black body temperature. Total energy absorbed is not a function of rotation speed.

Intuitively that doesn’t sound right. Ratio of time absorbing incoming radiation to the time emitting is constant regardless of rotation rate. Also, the total energy absorbed must equal the total energy emitted, which results in the black body temperature. Total energy absorbed is not a function of rotation speed.

I beg to differ since neither the Moon, nor the Earth are perfect bodies, and therefore do not behave in an ideal fashion, especially where there are lagged responses at work.

Contrary to that depicted in the K&T energy budget cartoon, the oceans absorb solar irradiance at depth, not at the surface, but radiate energy from the surface. The rate of rotation is material with such a set up since it takes time for the energy absorbed at depth to reach the surface from where the energy is radiated. Further, some of the incoming irradiance makes its way downward to depth and goes to heat the deep ocean, rather than simply returning to the surface. Given these processes, the pulsing of incoming irradiance becomes material.

Don;t forget that the rate of rotation of our planet is slowing, a day has slowed from about 4 hours to about 24 hours, and this is one contributory factor as to why the planet is cooling, Born hot, growing cold until such time as the sun begins to expand.

Richard
The surface temperature is a construct. It represents the average temperature of the surface of the object as a whole. Spinning or not spinning. The amount of energy going out per unit time period from whatever sized sphere you choose.
“as the speed of rotation tends towards infinity, the average surface temperature tends towards that of the warm side. ”
No, as the speed of rotation tends towards infinity the surface spends half the time on the dark side and half on the light side
Or to put it accurately the average surface temperature at any latitude becomes the average surface temperature at that latitude day and night.
Computing a GHG effect is not sensible as any atmosphere as we know it would have been tossed into space well before reaching any super fast rotation let alone infinity.
The energy source, the sun, is not producing any more energy to heat the earth. The faster rotatation merely distributes the same heat more evenly over our time perception of 24 hours.
It is distance from the heat source which determines how much an object can heat up by with no atmosphere.
With an atmosphere the amount of heat radiated back into space is the same, GHG selectively take up more of the radiation that would otherwise have hit the surface plus the back radiation in essence heating up that layer they are in. Our surface air temperature is not a true surface of the earth temperature, just an aberration of atmospheric physics.

GHG selectively take up more of the radiation that would otherwise have hit the surface plus the back radiation in essence heating up that layer they are in.

But that does not appear to be the case on Mars. As I note, if you actually count the number of molecules of CO2 in the Martian atmosphere, and count the actual number of molecules of CO2 in Earth’s atmosphere, there are 10 times as many molecules of CO2 in the Martian atmosphere.

So this begs the question, why are not all these extra CO2 molecules found in the Martian atmosphere actually “heating up that layer they are in”?

Perhaps you will answer why there does not appear to be any measurable GHE on Mars.

The heat in must equal the heat out. If it is not a blackbody then some of the heat is reflected, not absorbed and remitted.

You are stating a law that only applies to perfect absorbers/emitters, perfect blackbodies. This law does not apply to imperfect bodies and that is why Earth is never in equilibrium. The energy budget on this planet never balances, on any time scale, ie., from day to day, from month to month, from year to year etc. It is in constant flux and change, because this planet is not a perfect absorber/emitter; it is not a perfect blackbody.

The comment about energy radiating from the surface of the ocean only not from depth is also wrong. A trivial proof is that you can see the bottom of a swimming pool.

This demonstrates a deep misunderstanding by you. The reason we can see the bottom of a swimming pool, or in clear oceans, the sea bed at about 200 metres, is to do with the optical absorption of visible light in water. It is what I have mentioned earlier, that, contrary to the K&T energy budget cartoon, solar irradiance is not absorbed at the ocean surface, but rather at a depth of several metres through to a couple of hundred metres (albeit that most solar irradiance is absorbed within the about the top 2 to 15 metre depth range).

In contrast LWIR is almost entirely absorbed in the first 10 vertical microns of the ocean (about 1/5th the thickness of a human hair). In fact due to the omni-directional basis of DWLWIR, it means that over 60% of all DWLWIR is absorbed within about 4 microns (such that it cannot go to heat the oceans, and at most powers evaporation). .

We are very fortunate that the spectrum of solar irradiance is such that it is absorbed over such a large volume of water, since had it been absorbed like LWIR, in the top microns of the oceans, the oceans would have boiled off from the top down aeons ago.

It is impossible to have a water world with water on the surface and no atmosphere.

Essentially, we are trying to consider what temperature this planet would have if it had no atmosphere, or had an atmosphere comprising only of non radiative/non GHGs such as Nitrogen, Oxygen, Argon only. When carrying out such a thought experiment, it is necessary to keep materials as nearly the same as possible, so one would not replace the surface with carbon soot, or with high reflective quartz.

One could replace the oceans with some other material that has the same absorptive qualities as does water, ie., largely transparent to solar irradiance so that solar irradiance is absorbed at depth, and substantially opaque to LWIR such that all LWIR is absorbed within the top microns. One could give this notional material the same specific/latent heat characteristics as water, including that involved in a notional phase change etc. This is one of the fundamental components that give this planet a lag and which means that the planet is not a perfect absorber/prefect emitter, ie., not a perfect blackbody.

This planet is very different from the moon, not only in its speed of rotation but also in its constituent components such that one would not expect it to have the same temperature as the moon whether this be without an atmosphere, or with an atmosphere made exclusively from non radiative/non GHGs.

as the speed of rotation tends towards infinity the surface spends half the time on the dark side and half on the light side

Whilst that statement is true, and would be relevant if the planet was a perfect blackbody and was a perfect absorber and a perfect emitter, but it is not. Therein lies the issue.

Further, and materially, the speed of rotation impacts upon both oceanic currents, atmospheric currents and the Coriolis effect. If we had different oceanic currents (eg the slow down of the Gulf Stream and the shutting off thereto) or different atmospheric currents/jet streams, or the Coriolis effect was different, we would have different surface temperatures and hence a different figure for the so called GHE would emerge. If these currents and jets streams were different the amount and distribution of ice at the poles would be different which in turn would impact upon amongst other things the albedo of the planet.

Whilst we may have some grasp on the TOA position, ie., amount of energy received at TOA, and the amount of energy radiated at TOA, thereafter everything goes off the rails. We have no idea what the surface temperature of this planet is, there are NASA papers putting the surface temperature of this planet at about 9 degC and up to about 18 degC. Without accurately knowing the surface temperature of this planet, we cannot ascertain whether the so called GHE is 38degC, 35 degC, 33degC, 28 degc, 25 degC etc.

It is only by chance that we observe the temperatures that we see today. The SST that we observe today, is only partially the product of the radiative budget, but also due to physical processes plate tectonics, ocean basin topography, resultant oceanic currents (in 3 dimensions), and whether the cold temperatures of the deep oceans have come to back to bite.

If the oceans had the same temperature as that observed at SST throughout their volume, we would not have ice ages. But in ice ages the bulk ocean temperature (of about 4 deg C) comes back to bite. This is not in the first instance due to any change in the so called GHE.

If we were to look at this planet during the Holocene Optimum, or in an Ice Age we would come to a different collusion as to the value of the GHE. Don’t forget that this planet has entered Ice Ages with high levels of CO2. The paleo proxy record shows this planet to have warmed with falling levels of CO2 and to have cooled with rising levels of CO2, and without any material change in the TSI.

The fact that our planet is not a perfect blackbody and has lagged responses make it very difficult to get a proper handle on the so called GHE, and it is almost certainly the case that the speed of rotation of this planet has a significant impact on its temperature. This planet is very different to the Moon, and cannot be compared to it.

Re CO2 in mars atmosphere.
Even though there may be more molecules there is very little atmosphere. It is impossible for that little atmosphere to hold much temperature in it even if it is all CO2.
Comparing apples and oranges as per title.

It does not matter if the planet is a perfect black body or not. The heat in must equal the heat out. If it is not a blackbody then some of the heat is reflected, not absorbed and remitted.
All other things being equal the speed of rotation in sensible discussion cannot effect the fact if heat in heat out.

When not equal as you state ie currents change due to faster rotation you are now talking about a differen world to the one we live in.
Yes GHG affect would vary under different conditions. It also varies if you increase the amount of CO2. No one is claiming ECS is an invariant figure.

It is impossible to have a water world with water on the surface and no atmosphere.
The comment about energy radiating from the surface of the ocean only not from depth is also wrong. A trivial proof is that you can see the bottom of a swimming pool. Similarly some energy goes straight from the surface to space hence one can see the continents and oceans from space.

Let me present other facet of mean, median and mode in the sharing of river water among the riparian states [discussed in my 1993 book (available on line) – Incomplete Gamma Model – rainfall probability estimates].

The issue of dispute of Krishna River water sharing: The first Tribunal in 1970s used 78 years data set that was available to him at that time, namely 1894-95 to 1971-72 which was agreed by the three riparian states. Though Justice Brijesh Kumar Tribunal in 2013 has 114 years data series [1894-95 to 2007-08] but used selected data series of 1961-62 to 2007-08 of 47 years only. This was not agreed by Andhra Pradesh State [AP]. The average of 47 years data series is higher than 78 data series by 185 tmc ft.

The Tribunal put forth several subjective, unscientific and illogical arguments without giving scientific reasoning for doing so. The tribunal argued that “They [47 and 78 years data series] do not match hence cannot be integrated”; “Such increase [185 tmc ft] as reflected seems to be quite natural & obvious —“; “The longer the time series [114 years data series], however, greater the chance that it is neither stationary, consistent nor homogeneous”; and “we are of the opinion that 47 years length of the series should be considered sufficient to assess water availability of river. It more than fulfils the minimum requirement of IS Code —“.

The AP rainfall data series since 1871 presented a 132 year cyclic pattern in which 66 years each form part of below & above the average pattern in successive periods, the period prior to 1935 presents below the average pattern [in which 12 years presented excess rainfall and 24 years presented deficit rainfall] and the same is now started from 2001; and from 1935 to 2000 presents above the average pattern [in which 24 years presented excess rainfall and 12 years deficit rainfall]. In such scenarios truncated data sets presents misleading inferences.

The water availability data of 114 years follow the rainfall pattern. That is 78 years and 47 years data sets form part of below and above the average patterns only. Thus the average of 78 years data set is lower than the average of 47 years data set and as well under sine curve pattern [cyclic variation] they both form a continuum. In fact in the below the average period even delta area did not receive its share of 181.2 tmc ft of water even when all the major projects were not ready to use their share of water on many years – the Tribunal presented this data in its report. In the present below the average part, Nagarjunasagar reservoir hasn’t reached to its full capacity on many years.

The accuracy of the minimum expected amounts derived at a given probability level depends up on the representativeness of the data used and the degree of skewness in the data set. When a given data series with least skewness follow normal distribution. To get unbiased estimates of water availability at a given probability level, the data series thus must follow normal distribution. Then such data series are termed as stationary, consistent and homogeneous. The Tribunal did not apply this test. NOAA presented Atlantic storms [tropical storms & Hurricanes] since 1913- to date. The probability data peaks at September 10 between June 1 and November 30.
If the data set follows normal distribution then, the Mean coincides with the median [50% probability value]. In the three data sets of 47, 78 and 114 years, the Mean coincides respectively at 58%, 42% and 48% probability levels. This clearly suggest that 114 years data series is very close to normal distribution with the Mean at the 48% probability level; and the other two are following skewed distribution with 78 years data showing lesser skew over that of 47 years data series.

114 years data is the best data series for probability study over the other two with 47 years data set on the lower side [poor]. This discounts the inferences made by the Tribunal on longer data series and choosing shorter data series of 47 years for the Study.

The Tribunal distributed water at three probability levels among the three riparian states – Maharashtra, Karnataka & AP –, namely: 75%, 65% and 58% [the Mean] using probability curve built by plotting the lowest to the highest values using 47 years data set, a subset of 114 years data series. They are 2130, 2293 & 2578 tmc ft.

On the probability curve built using 114 years data set, for these three water limits the probability levels changed as 75% [fixed], 55% and 41.5% respectively. That means as for as the award is concerned 2578 TMCFT of water is available in 58% of the years but in reality it is available to AP in 41.5% of the years only as 114 years data series cover both better and poor rainfall periods. Similarly, 2293TMCFT of water is available to AP in 55% of the years only in reality but not in 65% of the years.

Though it is a technical issue, this was put in the hands of judiciary with unfettered powers and thus causing distress to one state with biased award to favour another state.

It seems the three data sets, of differing lengths, are incommensurable, should not be considered together, as they represent, in order pf length: the near complete precipitation climatic cycle, and two that represent partial cycles, high and low. Have I got this more or less correct?

Kip,
Was your choice of High Blood Pressure versus age a wise choice to illustrate a pure approach to statistics and numbers?
One of the first things to do when looking at a graph is to say “Y is dependent on X. Are the axes the right way around?” You graph supposes that the incidence of HBP is dependent on age. Is age itself an independent variable? Maybe in medical studies it is not, because older patients might have been exposed to drugs now not in use; and younger patients might have been treated with newer drugs that affect HBP, drugs not used on older patients because of their prior treatments being deemed enough. There are also effects of dietary changes, with young patients perhaps enjoying?? fewer deleterious foods that can affect blood pressure. And other lifestyle matters like alcohol, where younger people in general will have consumed less volume so far than older folk; and exercise, with young patients more exposed to the gym during their lifetime than older. Finally, the population of patients is bounded with ages from 0 to 110 or so years, with different numbers of patients in each age bin. Maybe there is a case to select survey patients so that age bins are equally populated, same number of 30-year-olds as 70-year-olds. I note that it is easy to devise arguments against what I am suggesting here, partly depending on what you wish to target with such a study. The overall point is that for demonstration of number factors like average, mean, median etc, there are probably many examples that have no or few complications on the X-axis. You have pointed to some such effects for the Y-axis in your essay.
A good deal of this type of study, as you well know, is concerned with the identification and quantification of extraneous variables. In a simple study they are absent, But not all studies can be simple.
Geoff

Geoff ==> In medical studies, researchers, unable to find specific CAUSES, look for “risk factors” — they are investigating common disease-states to find commonalities that might led them to discover causes or open avenues to treatment.
Certainly, AGE — particularly, advanced age — thus appears as a risk factor for a lot of disease. Cancers almost always have age as a major risk factor in this sense — and the cancer epidemic is often said to be a result of “more old people still alive” — the population is living longer.
The HBP example is simply meant to illustrate the concepts of Mean, Median, and Mode as common types of averages.

Geoff,
Fundamentally, you are pointing out that in a time-series all the interacting or confounding variables may not stay constant. One of the hallmarks of a good scientific laboratory experiment is keeping all variables constant save one. In the real world of observing ‘Nature,’ that is rarely possible. Thus, some assumptions need to be made, such as “Other changes besides the monitored parameter are negligible.” Unfortunately, that is rarely stated explicitly and is probably rarely true. That is one of the serious problems with climatology.

Loren ==> Yes, true, not everyone knows that. I am a California boy, so do know it, but don’t expect my readers to know. In our home, they are simply the Sierras. I have hiked Mount Whitney four times, three before I was old enough to drive, from both the west and the east.
Always good to see someone paying close attention!
Here in NY we have the Kaaterskill Creek — which means Kaaters Creek Creek (Kill already means creek, yet Kaaterskill Creek is the official name)

Yawrate ==> Apples and pears (and Quince and “asian” pears) are all what are known as POME fruits, members of the plant family Rosaceae, sub-family pomoideae. So, yes, they are similar but not identical. They are similar enough that one could compare the commonalities and discuss the differences.
Apples and Oranges and Bananas are so different that they defy comparison, they are incommensurable — their only common factors being that they are all classified botanically and in the in grocery as Fruits – two of them grow on trees.

Nice “prim-er,” but you left out an important “average,” the three-point estimate or the triangular probability distribution. To work around unavoidable prediction errors because of incorrect assumptions and limited data, petroleum scientists often present predictions as a range of expected values or best estimates.

A mathematically exact formula can be derived to calculate a best estimate from a continuous probability function. In the absence of a large data set of predictions, say, independent global mean temperature anomaly predictions, a good approximation of a best estimate can be calculated from a triangular distribution function (Fig 1). The implication is that the average value of an experiment repeated many times will converge to the best estimate as the number of experiments increases. For a mean global temperature estimate in 2100 (setting aside any “Fruit Salad” issues), the better the definition of the probability distribution, the better the best estimate.

For the triangular distribution, the area under the curve can be shown to equal the value of (A + B +C)/3, which is the probability weighted average of the function, that is, the expected value or best estimate for the parameter represented by the distribution. Note that for extremely skewed distributions, B would equal C.

A = A low predicted value (a value near the 2.5% percentile of a probability fun
C = A high predicted value (a value near the 97.5% percentile of a probability function.
B = The mode of the probability function (the most frequently occurring value of a probability function. For a normal distribution, mode = mean = best estimate = expected value.)

The relevance of a discussion of probability distributions is to focus attention on both tails of a probability function. Climate alarmists continue to debate how high the temperature will rise, which is only one-half of the problem. The probability of lower temperatures is always ignored as well as the likelihood that policies appropriate for the warming case would be diametrically opposite to those appropriate for the cooling case. Under this reality, the damage that would be done by acting based on the wrong premise, a warming or a cooling planet, nullifies the need to take any action until the science is right. If the complete probability function is appropriately considered, the right answer would be to get the science right before promulgating environmental policies.

Regarding the Fruit Salad analogy, I do not find the discussion convincing or, at least, not clear. Using a simple example to graphically illustrate the problem and the likely extent of error might helpful. If you accept that the “Butterfly effect” has merit, whether a sample is over asphalt or over plowed land does not matter. (“Butterfly effect:” A butterfly flapping its wings in South America can affect the weather in Central Park.) The shortcomings of climate science predictions are that the dataset is insufficient, and the physics is the wrong physics. The problem may be insoluble with Newtonian physics.

Mandatory Talking Point

The fundamental problem of the GCMs is not only the database. Many physicists do not accept the premise that global circulation models can adequately describe the earth and the solar system based on classical physics. President Rosenbaum at Caltech recently posited that nature cannot be modeled with classical physics but theoretically might be modeled with quantum physics. Climate models are driven by classical physics. Quantum physics modeling technology is not yet developed and may never be developed adequately to model earth processes.

Tom ==> “For a mean global temperature estimate in 2100 (setting aside any “Fruit Salad” issues), the better the definition of the probability distribution, the better the best estimate. ”
Strictly speaking, there is no possible probability distribution for a future value of an unknown system. There is no way to determine the probabilities from existing data — we cannot predict the future.
I can see that one could take a triangular average of the “predicted” values — but they have no physical meaning. — they are, in effect, imaginary.

I’ll spend some time on your explanation — it is applicable to known probability distribution functions from real data.

Addendum ==> By “unknown system” I mean that the functions and the mechanisms of the system under consideration (Earth’s climate system creating “mean global temperature” ) is not well enough understood to predict effects far in the future from hypothesized changes in parameters.
Besides that immense problem, there is Chaos (which see my series here, here and here.)

The concept is not to predict a value but to predict a reasonable range of values in which the actual value will lie. The goal would be to understand boundary conditions for a prediction and how more data and model changes might narrow the range. My working paper referenced in the comment is a simple example using temperature data. I would be interested in critical replies and ideas. So far, my colleagues have mostly declined to comment. I am not presuming this method is suitable for long-term predictions. I am considering a thought experiment to show that understanding of the climate system might not be required to predict future temperatures. Not ready for prime time.

Tom ==> Thanks for not taking offense — there are so many in Climate Science (and economics) that feel that their maths and stats and trends can actually predict or project the future. So that one sentence attracted my attention.
As I say “I’ll spend some time on your explanation — it is applicable to known probability distribution functions from real data.”

I don’t fundamentally disagree with your statement: “The concept is not to predict a value but to predict a reasonable range of values in which the actual value will lie.” However, you haven’t provided a definition of “reasonable.” If the range is so large that the future value may lie between catastrophe and a big yawn, then it isn’t very useful!

I’d suggest that an initial definition would be a range that has utility in preparing for future changes. However, if an attempt is made to predict a single value, provided with a standard deviation, either explicitly, or implied by the number of significant figures, then we have gone a long way towards meeting your suggestion. The standard deviation is something that one rarely sees in climatology data.

Thank you for an important post. It cannot be emphasized enough that good science cannot come from poor statistics.

I will take issue with both assertions that “The global temperature exists. It has a precise physical meaning.” Neither is true. If Mosher asserts otherwise, perhaps he will be good enough to tell us what it. As I have said, if a hot coal falls out of my wood stove it is not relevant that the average temperature in
my room is suddenly 500 oF.

It has been asserted that CO2 changes atmospheric energy. If so it would be useful to measure this energy. The measure of energy density of air is enthalpy which is an intensive property. A change in enthalpy of air cannot attributed solely to its temperature, but must include other factors, most noteably water vapor content. Temperature alone measures the energy of the atmosphere about as well as your checking account measures your net worth — relevant, but not the whole story.

As for the “surface waters” study, I gave up reading when I found they did not isolate “dew point”, which in my limited nautical experience is highly important for changing surface water temperature.

While the concept of average can be useful in characterizing variability, many will be surprised to know that “average” and/or “standard deviation” are not needed for comparing samples from variable processes. Much useful statistical work can be done using variations of the Kolmogorov-Smirnov statistical tests, which require only the empirical cumulative distributions. If you are faced with handling highly variable data I recommend understanding how this family of tests functions. It would be interesting to apply these statistics to the Reddy case noted earlier. (I will note that a finite cumulative distribution cannot be differentiated.)

The important point is that I claim that the standard deviation for global temperatures is of the order of 10’s of degrees Fahrenheit! So, while Mosher claims that the average global temperature has a precise value, he overlooks the fact that the range is so large that the number has little practical value. Probably what is more important is whether the claimed changes in climate will increase or decrease (or have no effect!) the range in temperatures. If the Arctic region is warming more rapidly than the global average, then it appears that the future will have a decreased range in temperatures.

Incidentally, it just occurred to me that it is conceivable, with a skewed temperature distribution such as Earth apparently has, that the mean would increase if the annual lows increased, while there might be no change in either the mode or the highest temperatures. Calculating a mean alone is not particularly informative for Earth temperatures. The frequency distribution over time is more informative and the Kolmogorov-Smirnov tests suggested by 4Kx3 might be useful in assessing those changes.

My thanks to all of you reading this essay and taking part in the discussion.

Some interesting comments, some good examples of Fruit Salad averages from differing fields of endeavor.

I appreciate those who pay attention to individual words and spot my typos — my usual editor has been off-duty for my last few essays, leaving me to my own devices.

The most important take-home from this essay is being able to recognize that when someone puts forward a Fruit Salad average claiming that it proves or shows this-or-that, in almost every case (leaving a narrow way out for the one-in-a-million possibility) it does no such thing — averages of incommensurable data give results that are generally illogical or non-physical.

I hope some of you look forward to “The Laws of Averages: Part 2 – A Beam of Darkness”.

Though the idea of a global average temperature (GAT) could have some meaning, I don’t believe the current incarnations have any whatsoever. In the first place, as far as I can tell, the GISS GAT doesn’t use even one measured temperature in its calculation. Each of the 8,000 equal size grid plates (each almost exactly the size of West Virginia) is assigned a temperature anomaly based on what it would have seen based on a temperature measurement no more than 1,250 km away, according to some obscure statistical correlation. So a temperature measurement that has an error of no less than +/- 0.5 C is used, after “adjustment” for things such as time of day and location, is used to estimate the temperature of a place 1,250 km away, and all of the estimated temperatures are averaged to find a result that is quoted to two decimal places (when no decimal places are warranted).

Let’s take the emotion out of it, and look at a different situation. Let’s say that every day, 3,000 people around the world put a target up on a tree, stand back 25 yards, and fire one pistol shot at the target. Then they measure the distance from the exact center to the bullet hole, and report it to a central repository. That central repository takes all of the reported measurements,adds them together, divides by the total number of measurements, and posts the results as the “global average shooting accuracy.” It is sort of a meaningful number, but has the following characteristics: since it is the average of individual measurements having no connection whatsoever, no amount of averaging can increase the “accuracy” of the number. Accuracy is the difference between the measured and “actual” value. There isn’t, and cannot be, any “actual” value in this case, even though they’re all miss distances.

Further, the existence of measurement bias would call into question even the “accuracy” of the average (and that can have an accuracy). We don’t know whether people are using accurate rulers, or whether they are measuring from the exact target center to the center of the bullet hole or to its edge. And if to the center of the bullet hole, how much error is introduced there (partly a function of bullet caliber, which is not constrained). Any bias won’t be averaged out, and will increase the “error” in a number which really has no meaning to begin with.

Michael,
And what is the precision of an anomaly that is obtained by subtracting a number, with a standard deviation that is 10s of degrees, from a number, say an annual average, that has a standard deviation of a similar magnitude? Can one really believe that two very imprecise numbers can produce a result that that is two orders of magnitude more precise than any of the original raw temperature measurements?

That was a pleasure to read. A great way to simplify what can be very complicated – statistics. We see the same “apples and oranges” issue in mining in trying to quantify the average grade of an orebody where there are different sampling techniques used. My example maybe a bit closer to Granny Smith apples and Red Delicious apples, but it is well known that data sets derived from different sample types, or even different ages, need to be treated with caution. More often than not these data sets will be mixed, because of cost issues, and a good approximation of average grade is derived, but that takes a lot of skill.

Some reading to explore this further:
1. Savage, S. L., & Markowitz, H. M. (2009). The flaw of averages: Why we underestimate risk in the face of uncertainty. John Wiley & Sons.
2. Google “Doksum shift plot” or function. The change in mean is only meaningful when the shift is uniform. (That’s a technical version of the fruit salad function) (Strict homogeniety is not as important when you have before and after measures on each object. Heterogeniety makes it more informative, as in “how stable is this effect across strata?”)
3. A related variation is the Bland-Altman plot, to show how the shift changes with location. aka a Tukey sum-difference plot of CDF percentiles.
4. Then there are QQ plots, PP plots, parallel coordinate plots and other variations on the same theme.